Digital Signal Processing
BEC505
Chapter 1: Introduction
What is a Signal?
Anything which carries information is a signal. e.g. human voice, chirping of birds, smoke signals,
gestures (sign language), fragrances of the flowers.
Many of our body functions are regulated by chemical signals, blind people use sense of touch. Bees
communicate by their dancing pattern.
Modern high speed signals are: voltage changer in a telephone wire, the electromagnetic field
emanating from a transmitting antenna,variation of light intensity in an optical fiber.
Thus we see that there is an almost endless variety of signals and a large number of ways in which
signals are carried from on place to another place.
Signals: The Mathematical Way
A signal is a real (or complex) valued function of one or more real variable(s).When the function
depends on a single variable, the signal is said to be one-dimensional and when the function depends on
two or more variables, the signal is said to be multidimensional.
Examples of a one dimensional signal: A speech signal, daily maximum temperature, annual rainfall at a
place
An example of a two dimensional signal: An image is a two dimensional signal, vertical and horizontal
coordinates representing the two dimensions.
Four Dimensions: Our physical world is four dimensional(three spatial and one temporal).
What is Signal processing?
Processing means operating in some fashion on a signal to extract some useful information e.g. we use
our ears as input device and then auditory pathways in the brain to extract the information. The signal is
processed by a system. In the example mentioned above the system is biological in nature.
The signal processor may be an electronic system, a mechanical system or even it might be a computer
program.
Analog versus digital signal processing
The signal processing operations involved in many applications like communication systems, control
systems, instrumentation, biomedical signal processing etc can be implemented in two different ways
Analog or continuous time method
Digital or discrete time method..
Analog signal processing
Uses analog circuit elements such as resistors, capacitors, transistors, diodes etc
Based on natural ability of the analog system to solve differential equations that describe a physical
system
The solutions are obtained in real time...
Digital signal processing
The word digital in digital signal processing means that the processing is done either by a digital
hardware or by a digital computer.
Relies on numerical calculations
The method may or may not give results in real time..
The advantages of digital approach over analog approach
Flexibility: Same hardware can be used to do various kind of signal pr
case of analog signal processing one has to design a system for each kind of operation
Repeatability: The same signal processing operation can be repeated again and again giving same
results, while in analog systems there ma
supply voltage.
The choice of choosing between analog or digital signal processing depends on the application. One has
to compare design time,size and thecost of the implementation.
Classification of signals
We use the term signal to mean a real or complex valued function of real variable(s) and denote the
signal by x(t)
The variable t is called independent variable and the value x of t as dependent variable.
When t takes a vales in a countable set t
t ε {0, T, 2T, 3T, 4T,...}
t ε {....-1, 0 ,1,...}
t ε {1/2, 3/2, 5/2, 7/2,...}
For convenience of presentation we use the notation x[n] to denote dis
dependent and independent variables take values in countable sets (two sets can be quite different) the
signal is called Digital Signal.
When both the dependent and independent variable take value in continous set interval,
called an Analog Signal.
Notation:
When we write x(t) it has two meanings. One is value of x at time t and the other is the pairs (x(t), t)
allowable value of t. By signal we mean the second interpretation.
Notation for continous time signal
{x(t)} denotes the continuous time signal. Here {x(t)} is short notation for {x(t), t ε I } where I is the set in
which t takes the value.
Notation for discrete time signal
Similarly for discrete time signal we will use the notation {x(t)}, where {x(t)}
Note that in {x(t)} and {x[n]} are dummy variables ie. {x[n]} and {x[t]} refer to the same signal. Some
books use the notation x [.] to denote {x[n]} and x[n] to denote value of x at time n.
{x(t)} refers to the whole waveform,while x[n] refers to a particular value.
Most of the books do not make this distinction clean and use x[n] to denote signal and x[n0] to denote a
particular value.
Discrete Time Signal Processing and Digital Signal Processing
When we use digital computers to do processing we are doing digital signal processing. But most of the
theory is for discrete time signal processing where dependent variable generally is continuous. This is
because of the mathematical simplicity of discrete time signal proces
to implement this as closely as possible. Thus what we study is mostly discrete time signal processing
and what is really implemented is digital signal processing.
Elementary Signals
There are several elementary signals that occur prominently in the study of digital signals and digital
signal processing.
(a) UNIT SAMPLE SEQUENCE:
The advantages of digital approach over analog approach
Flexibility: Same hardware can be used to do various kind of signal processing operation,while in the
case of analog signal processing one has to design a system for each kind of operation
Repeatability: The same signal processing operation can be repeated again and again giving same
results, while in analog systems there may be parameter variation due to change in temperature or
The choice of choosing between analog or digital signal processing depends on the application. One has
to compare design time,size and thecost of the implementation.
We use the term signal to mean a real or complex valued function of real variable(s) and denote the
The variable t is called independent variable and the value x of t as dependent variable.
When t takes a vales in a countable set the signal is called a discrete time signal. For example
1, 0 ,1,...}
For convenience of presentation we use the notation x[n] to denote discrete time signal. When both the
dependent and independent variables take values in countable sets (two sets can be quite different) the
When both the dependent and independent variable take value in continous set interval,
When we write x(t) it has two meanings. One is value of x at time t and the other is the pairs (x(t), t)
allowable value of t. By signal we mean the second interpretation.
{x(t)} denotes the continuous time signal. Here {x(t)} is short notation for {x(t), t ε I } where I is the set in
Similarly for discrete time signal we will use the notation {x(t)}, where {x(t)} is short for {x(t), n ε I }.
Note that in {x(t)} and {x[n]} are dummy variables ie. {x[n]} and {x[t]} refer to the same signal. Some
books use the notation x [.] to denote {x[n]} and x[n] to denote value of x at time n.
rm,while x[n] refers to a particular value.
Most of the books do not make this distinction clean and use x[n] to denote signal and x[n0] to denote a
Discrete Time Signal Processing and Digital Signal Processing
computers to do processing we are doing digital signal processing. But most of the
theory is for discrete time signal processing where dependent variable generally is continuous. This is
because of the mathematical simplicity of discrete time signal processing. Digital Signal Processing tries
to implement this as closely as possible. Thus what we study is mostly discrete time signal processing
and what is really implemented is digital signal processing.
signals that occur prominently in the study of digital signals and digital
ocessing operation,while in the
Repeatability: The same signal processing operation can be repeated again and again giving same
y be parameter variation due to change in temperature or
The choice of choosing between analog or digital signal processing depends on the application. One has
We use the term signal to mean a real or complex valued function of real variable(s) and denote the
The variable t is called independent variable and the value x of t as dependent variable.
he signal is called a discrete time signal. For example
crete time signal. When both the
dependent and independent variables take values in countable sets (two sets can be quite different) the
When both the dependent and independent variable take value in continous set interval, the signal is
When we write x(t) it has two meanings. One is value of x at time t and the other is the pairs (x(t), t)
{x(t)} denotes the continuous time signal. Here {x(t)} is short notation for {x(t), t ε I } where I is the set in
is short for {x(t), n ε I }.
Note that in {x(t)} and {x[n]} are dummy variables ie. {x[n]} and {x[t]} refer to the same signal. Some
Most of the books do not make this distinction clean and use x[n] to denote signal and x[n0] to denote a
computers to do processing we are doing digital signal processing. But most of the
theory is for discrete time signal processing where dependent variable generally is continuous. This is
sing. Digital Signal Processing tries
to implement this as closely as possible. Thus what we study is mostly discrete time signal processing
signals that occur prominently in the study of digital signals and digital
Defined by
Graphically this is as shown below.
Unit sample sequence is also known as
This plays role akin to the impulse function
is purely a mathematical construct while in discrete time we can actually generate the
sequence.
(b) UNIT STEP SEQUENCE:
Defined by :
Graphically this is as shown below
(c) EXPONENTIALSEQUENCE:
The complex exponential signal or sequence {x[n]}
where C and α are, in general, complex numbers.
Note that by writing α = eβ , we can write the exponential sequence as
Real exponential signals:
: If C and are real, we can have one of the several type of behavior illustrated below
Unit sample sequence is also known as impulse sequence.
role akin to the impulse function of continous time. The continues time impulse
is purely a mathematical construct while in discrete time we can actually generate the
The complex exponential signal or sequence {x[n]} is defined by x[n] = C αn
and α are, in general, complex numbers.
, we can write the exponential sequence as x[n] = c eβn
are real, we can have one of the several type of behavior illustrated below
of continous time. The continues time impulse
is purely a mathematical construct while in discrete time we can actually generate the impulse
are real, we can have one of the several type of behavior illustrated below
For |α| > 1
|α| < 1
For α > 1
α < 1 sign of terms in {x[n]}
(d)SINUSOIDAL SIGNAL:
The sinusoidal signal {x[n]} is defined by
Euler's relation allows us to relate complex exponentials and sinusoids as
and
The general discrete time complex exponential can be written in terms of real exponential and
sinusiodal signals.
Specifically if we write C and α in polar form
Thus for |α| = 1 , the real and imaginary part
|α| < 1, they correspond to sinusoidal sequence multiplied by a decaying exponential,
|α| > 1 , they correspond to sinusiodal sequence multiplied by a growing exponential
Generating Signals with MATLAB
MATLAB, acronym for MATrix LABoratory has become a very popular software environment for
complex based study of signals and systems. Here we give some sample programmes
elementary signals discussed above. For details one should consider MATLAB manual or read help
files.
In MATLAB, ones(M,N) is an M-
zeros. We may use those two matrices to generate
magnitude of the signals grows exponentially,
|α| < 1 It is decaying exponential.
all terms of {x[n]} have same sign,
sign of terms in {x[n]} alternates.
is defined by
Euler's relation allows us to relate complex exponentials and sinusoids as
The general discrete time complex exponential can be written in terms of real exponential and
and α in polar form and then
, the real and imaginary parts of a complex exponential sequence are sinusoidal.
|α| < 1, they correspond to sinusoidal sequence multiplied by a decaying exponential,
|α| > 1 , they correspond to sinusiodal sequence multiplied by a growing exponential
MATLAB, acronym for MATrix LABoratory has become a very popular software environment for
complex based study of signals and systems. Here we give some sample programmes
elementary signals discussed above. For details one should consider MATLAB manual or read help
-by-N matrix of ones, and zeros(M,N) is an M-
We may use those two matrices to generate impulse and step sequence.
magnitude of the signals grows exponentially,
It is decaying exponential.
all terms of {x[n]} have same sign,
The general discrete time complex exponential can be written in terms of real exponential and
s of a complex exponential sequence are sinusoidal.
|α| < 1, they correspond to sinusoidal sequence multiplied by a decaying exponential,
|α| > 1 , they correspond to sinusiodal sequence multiplied by a growing exponential.
MATLAB, acronym for MATrix LABoratory has become a very popular software environment for
to generate the
elementary signals discussed above. For details one should consider MATLAB manual or read help
by-N matrix of
The following is a program to generate and display impulse sequence.
Here >> indicates the MATLAB prompt to type in a command,
vector xas a discrete time signal at time values defined by n. One can add title and lable the axes by
suitable commands. To generate step sequence we can use the following program
We can use the following program to generate real exponential sequence
Note that, in this program, the base alpha is a scalar but the exponent is a vector, hence use of the
operator to denote element-by
Recap
In last lecture you have learnt the following
Signals are functions of one or more
Systems are physical models which gives out an output signal in response to an input signals.
Trying to identify real-life examples as models of signals and systems, would help us in
understanding the subject better.
Objectives
In this lecture you will learn the following
The following is a program to generate and display impulse sequence.
indicates the MATLAB prompt to type in a command, stem(n,x) depicts the data contained in
a discrete time signal at time values defined by n. One can add title and lable the axes by
suitable commands. To generate step sequence we can use the following program
We can use the following program to generate real exponential sequence
Note that, in this program, the base alpha is a scalar but the exponent is a vector, hence use of the
by-element power.
In last lecture you have learnt the following
Signals are functions of one or more independent variables.
Systems are physical models which gives out an output signal in response to an input signals.
life examples as models of signals and systems, would help us in
understanding the subject better.
In this lecture you will learn the following
depicts the data contained in
a discrete time signal at time values defined by n. One can add title and lable the axes by
Note that, in this program, the base alpha is a scalar but the exponent is a vector, hence use of the
Systems are physical models which gives out an output signal in response to an input signals.
life examples as models of signals and systems, would help us in
In this chapter we will learn some of the operations performed on the sequences.
Sequence
Addition
Scalar Multiplication
Sequence Multiplication
Shifting
Reflection
we will learn some of the properties of signals.
Energy of a signal
Power of a signal
Periodicity of signals
Even and Odd signals
Periodicity property of sinusoidal signals
Sequence addition:Let {x[n]} and {y[n]}be two sequences. The sequence addition is defined as term by
term addition. Let {z[n]} be the resulting sequence
where each term
{x[n]} + {y[n]} = {x[n] + y[n]}
Scalar multiplication:Let a be a scalar. We will take
signals, and take to be a complex number if
otherwise stated we will consider complex valued sequences. Let the resulting sequence be denoted
by {w[n]}
is defined by
each term is multiplied by a We will use the notation
Note: If we take the set of all sequences and define these two operations as addition and scalar
multiplication they satisfy all the properties of a linear
Sequence multiplication:
Let {x[n]} and {y[n]} be two sequences, and {z[n]} be resulting sequence
where
The notation used for this will be
Now we consider some operations based on independent variable
Shifting:
This is also known as translation. Let us shift a sequence
be{y[n]}
we will learn some of the operations performed on the sequences.
will learn some of the properties of signals.
Periodicity property of sinusoidal signals
{x[n]} and {y[n]}be two sequences. The sequence addition is defined as term by
term addition. Let {z[n]} be the resulting sequence
{z[n]} = {x[n]} + {y[n]}
z[n] = x[n] + y[n]We will use the following notation
{x[n]} + {y[n]} = {x[n] + y[n]}
be a scalar. We will take a to be real if we consider only the real valued
to be a complex number if we are considering complex valued sequence. Unless
otherwise stated we will consider complex valued sequences. Let the resulting sequence be denoted
{w[n]} = a {x[n]}
w[n] = ax[n]
We will use the notation
a {w[n]} = {aw[n]}
Note: If we take the set of all sequences and define these two operations as addition and scalar
multiplication they satisfy all the properties of a linear vector space.
{y[n]} be two sequences, and {z[n]} be resulting sequence
{z[n]} = {x[n]}{y[n]}
z[n] = x[n] y[n]
The notation used for this will be {x[n]} {y[n]} = {x[n] y[n]}
Now we consider some operations based on independent variable n.
known as translation. Let us shift a sequence {x[n]} by n0 units, and the resulting sequence
we will learn some of the operations performed on the sequences.
{x[n]} and {y[n]}be two sequences. The sequence addition is defined as term by
lowing notation
to be real if we consider only the real valued
we are considering complex valued sequence. Unless
otherwise stated we will consider complex valued sequences. Let the resulting sequence be denoted
Note: If we take the set of all sequences and define these two operations as addition and scalar
units, and the resulting sequence
where is the operation of shifting the sequence right by n
x[n - n0]. We will use short notation
Figure below show some examples of shifting.
{x[n]}
{x[n-2]}
{x [n+1]}
Reflection:
Let {x[n]} be the original sequence, and {y[n]} be reflected sequence, then y[n] is defined by
{x[n]}
is the operation of shifting the sequence right by n0 unit. The terms are defined
]. We will use short notation {x[n - n0]} to denote shift by n0.
Figure below show some examples of shifting.
Consider the figure to the left.
A negative value of n0 means shift
A positive value of n0 means shift towards left.
Let {x[n]} be the original sequence, and {y[n]} be reflected sequence, then y[n] is defined by
y[n] = x[-n]
unit. The terms are defined by y[n] =
means shift towards right.
means shift towards left.
Let {x[n]} be the original sequence, and {y[n]} be reflected sequence, then y[n] is defined by
We will denote this by {x[-n]}
When we have complex valued signals, sometimes we reflect and do the complex conjugation, ie, y[n]
is defined by y[n] = x*[-n], where * denotes complex conjugation. This sequence will be denoted by
{x*[-n]}.
We will learn about more complex operations later on. Some of these operations commute, ie. if we
apply two operations we can interchange their order and some do not commute. For example scalar
multiplication and reflection commute.
Then v[n] = z[n] for all n. Shifting and scaling do not commute.
{x[n]} {y[n]} = {x[n-1} {z[n]} = {y[-n]}
{x[n]} {w[n]} = {x[
We can combine many of these operations in one step, for example {y[n]} may be defined as
[3-n].
Some Properties of signals
Energy of a Signal:
The total energy of a signal {x[n]} is defined by
A signal is referred to as an energy signal, if and only if the total energy of the signal E
Power of a signal:If {x[n]} is a signal whose energy is not finite, we define power of the signal as
A signal is referred to as a power signal if the power P
An energy signal has a zero power and a power signal has infinite energy. There are signals which are
neither energy signals nor power signals. For example {x[n]} defined by
power or energy.
Periodic Signals:
An important class of signals that we encounter frequently is the class of periodic signals. We say that
a signal {x[n] is periodic with period N, where N is a positive integer, if the signal is unchanged by the
time shift of N ie.,
or x[n] = x[ n + N ] for all n.
Since {x[n]} is same as {x[n+N]} , it is also periodic so we get
{x[n]} = {x[n+N]} = {x[n+N+N]}
Generalizing this we get {x[n]} = {x[n+kN]}, where k is a positive integer. From
periodic with 2N, 3N,... The fundamental period N
is periodic.
{w[n]} = {x[-n]} {u[n]} = {w[n-1]}
We can combine many of these operations in one step, for example {y[n]} may be defined as
The total energy of a signal {x[n]} is defined by
to as an energy signal, if and only if the total energy of the signal Ex
If {x[n]} is a signal whose energy is not finite, we define power of the signal as
A signal is referred to as a power signal if the power Px satisfies the condition
An energy signal has a zero power and a power signal has infinite energy. There are signals which are
neither energy signals nor power signals. For example {x[n]} defined by x[n] = n does not have finite
An important class of signals that we encounter frequently is the class of periodic signals. We say that
a signal {x[n] is periodic with period N, where N is a positive integer, if the signal is unchanged by the
{x[n]} = {x[n + N]}
, it is also periodic so we get
{x[n]} = {x[n+N]} = {x[n+N+N]} = {x[n+2N]}
Generalizing this we get {x[n]} = {x[n+kN]}, where k is a positive integer. From this we see that
periodic with 2N, 3N,... The fundamental period N0 is the smallest positive value N for which the signal
We can combine many of these operations in one step, for example {y[n]} may be defined as y[n] = 2x
x is finite.
If {x[n]} is a signal whose energy is not finite, we define power of the signal as
An energy signal has a zero power and a power signal has infinite energy. There are signals which are
does not have finite
An important class of signals that we encounter frequently is the class of periodic signals. We say that
a signal {x[n] is periodic with period N, where N is a positive integer, if the signal is unchanged by the
this we see that {x[n]} is
is the smallest positive value N for which the signal
The signal illustrated below is periodic with fundamental period N
By change of variable we can write
we see that
for all integer values of k positive, negative or zero. By definition, period of a signal is always a positive
integer N.
Except for a all zero signal all periodic signals have infinite energy. They may have finite power. Let
{x[n]} be periodic with period N, then the power P
where k is largest integer such that
be same for all terms. We see that k is approximately equal to
large M we get 2M/N terms and limit
Even and odd signals:
A real valued signal {x[n]} is referred to as an even signal if it is identical to its time reversed
counterpart ie, if
{x[n]} = {x[-n]}
A real signal is referred to as an odd signal if
An odd signal has value 0 at n = 0 as
Given any real valued signal {x[n]} we can write it as a sum of an even signal and an odd signal.
The signal illustrated below is periodic with fundamental period N0 = 4
By change of variable we can write {x[n]} = {x[n+N]} as {x[m-N]} = {x[m]} and then arguing as before,
{x[n]} = {x[n+kN]},
for all integer values of k positive, negative or zero. By definition, period of a signal is always a positive
Except for a all zero signal all periodic signals have infinite energy. They may have finite power. Let
{x[n]} be periodic with period N, then the power Px is given by
is largest integer such that kN -1 ≤ M. Since the signal is periodic, sum over one period will
be same for all terms. We see that k is approximately equal to M/N (it is integer part of this) and for
terms and limit 2M/(2M +1) as M goes to infinite is one we get
is referred to as an even signal if it is identical to its time reversed
is referred to as an odd signal if
{x[n]} = {-x[-n]}
An odd signal has value 0 at n = 0 as x[0] = -x[n] = - x[0]
{x[n]} we can write it as a sum of an even signal and an odd signal.
N]} = {x[m]} and then arguing as before,
for all integer values of k positive, negative or zero. By definition, period of a signal is always a positive
Except for a all zero signal all periodic signals have infinite energy. They may have finite power. Let
≤ M. Since the signal is periodic, sum over one period will
M/N (it is integer part of this) and for
as M goes to infinite is one we get
is referred to as an even signal if it is identical to its time reversed
{x[n]} we can write it as a sum of an even signal and an odd signal.
Consider the signals
Ev ({x[n]}) = {xe[n]} = {1/2 (x[n] + x[
and Od ({x[n]}) = {xo[n]} = {1/2(x[n]
We can see easily that
{x[n]} = {xe[n]} + {xo
The signal {xe[n]} is called the even part of
signal. Similarly, {xo[n]} is called the odd part of {x[n]} and is an odd signal. When we have complex
valued signals we use a slightly different terminology. A complex valued signal {x[n]} is refer
a conjugate symmetric signal if
{x[n]} = {x*[-n]}
where x* refers to the complex conjugate of x. Here we do reflection and complex conjugation. If
{x[n]} is real valued this is same as an even signal.
A complex signal {x[n]} is referred to as a conjugate antisymmetric signal if
We can express any complex valued signal as sum conjugate symmetric and conjugate antisymmetric
signals. We use notation similar to above
Ev({x[n]}) = {xe[n]} = {1/2(x[n] + x*[
and Od ({[n]}) = {xo[n]} = {1/2(x[n]
then {x[n]} = {xe[n]} + {xo[n]}
We can see easily that {xe[n]} is conjugate symmetric signal and
signal. These definitions reduce to even and odd signals in case signals takes only real values.
Periodicity properties of sinusoidal signals:
Let us consider the signal. We see that if we replace
the signal with frequency
continuous time signal
time we need to consider frequency interval of length 2π only. As we increase
oscillates more and more rapidly. But if we further increase frequency from π to 2π the rate of
oscillations decreases. This can be seen easily by plotting signal
The signal is not periodic for every value of. For the signal to be periodic with period N
we should have
that is should be some multiple of 2π.
or
Thus signal is periodic if and only if
Above observations also hold for complex exponential signal
Discrete-Time Systems:
Consider the signals
[n]} = {1/2 (x[n] + x[-n])}
[n]} = {1/2(x[n] -x [-n])}
o[n]}
[n]} is called the even part of {x[n]}. We can verify very easily that {xe
[n]} is called the odd part of {x[n]} and is an odd signal. When we have complex
valued signals we use a slightly different terminology. A complex valued signal {x[n]} is refer
where x* refers to the complex conjugate of x. Here we do reflection and complex conjugation. If
{x[n]} is real valued this is same as an even signal.
signal {x[n]} is referred to as a conjugate antisymmetric signal if
{x[n]} = {-x*[-n]}
We can express any complex valued signal as sum conjugate symmetric and conjugate antisymmetric
signals. We use notation similar to above
[n]} = {1/2(x[n] + x*[-n])}
[n]} = {1/2(x[n] - x*[-n])}
[n]} is conjugate symmetric signal and {xo[n]} is conjugate antisymmetric
signal. These definitions reduce to even and odd signals in case signals takes only real values.
Periodicity properties of sinusoidal signals:
Let us consider the signal. We see that if we replace by we get the same signal. In fact
and so on. This situation is quite different from
where each frequency is different. Thus in discrete
time we need to consider frequency interval of length 2π only. As we increase
apidly. But if we further increase frequency from π to 2π the rate of
oscillations decreases. This can be seen easily by plotting signal for several values of.
is not periodic for every value of. For the signal to be periodic with period N
should be some multiple of 2π.
is periodic if and only if is a rational number.
Above observations also hold for complex exponential signal
e[n]} is an even
[n]} is called the odd part of {x[n]} and is an odd signal. When we have complex
valued signals we use a slightly different terminology. A complex valued signal {x[n]} is referred to as
where x* refers to the complex conjugate of x. Here we do reflection and complex conjugation. If
{x[n]} is real valued this is same as an even signal.
We can express any complex valued signal as sum conjugate symmetric and conjugate antisymmetric
[n]} is conjugate antisymmetric
signal. These definitions reduce to even and odd signals in case signals takes only real values.
we get the same signal. In fact
and so on. This situation is quite different from
where each frequency is different. Thus in discrete
to π signal
apidly. But if we further increase frequency from π to 2π the rate of
for several values of.
is not periodic for every value of. For the signal to be periodic with period N > 0,
A discrete-time system can be thought of as a transformation or operator that maps an input sequence
{x[n]} to an output sequence {yk[n]}
By placing various conditions on T(
Basic System Properties
• Systems with or without memory:
• Invertibility
• Causality
• Stability
• Time invariance
• Linearity
Systems with or without memory:
A system is said to be memoryless if the output for each value of the independent variable at a given
time n depends only on the input value at time n. For example system specified by the relationship
is memoryless. A particularly simple memoryless
In general we can write input-output relationship for memoryless system as
Not all systems are memoryless. A simple example of system with memory is a delay defined by
A system with memory retains or stores information about input values at times other than the
current input value.
Invertibility:
A system is said to be invertible if the input signal {x[n]} can be recovered from the output signal {y
For this to be true two different input signals should produce two different outputs. If some different
input signal produce same output signal then by processing output we can not say which input
produced the output.
Example of an invertible system is
then
Example if a non-invertible system is
That is the system produces an all zero sequence for any input sequence. Since every input seque
gives all zero sequence, we can not find out which input produced the output.
The system which produces the sequence {x[n]} from sequence {y
communication system, decoder is an inverse of the encoder.
time system can be thought of as a transformation or operator that maps an input sequence
[n]}
By placing various conditions on T(.) we can define different classes of systems.
Systems with or without memory:
if the output for each value of the independent variable at a given
time n depends only on the input value at time n. For example system specified by the relationship
y[n] = cos (x[n]) + 3
is memoryless. A particularly simple memoryless system is the identity system defined by
y[n] = x[n]
output relationship for memoryless system as
y[n] = g(x[n])
Not all systems are memoryless. A simple example of system with memory is a delay defined by
y[n] = x[n-1]
A system with memory retains or stores information about input values at times other than the
A system is said to be invertible if the input signal {x[n]} can be recovered from the output signal {y
For this to be true two different input signals should produce two different outputs. If some different
input signal produce same output signal then by processing output we can not say which input
invertible system is
That is the system produces an all zero sequence for any input sequence. Since every input seque
gives all zero sequence, we can not find out which input produced the output.
The system which produces the sequence {x[n]} from sequence {yk[n]} is called the inverse system. In
communication system, decoder is an inverse of the encoder.
time system can be thought of as a transformation or operator that maps an input sequence
if the output for each value of the independent variable at a given
time n depends only on the input value at time n. For example system specified by the relationship
system is the identity system defined by
Not all systems are memoryless. A simple example of system with memory is a delay defined by
A system with memory retains or stores information about input values at times other than the
A system is said to be invertible if the input signal {x[n]} can be recovered from the output signal {yk[n]}.
For this to be true two different input signals should produce two different outputs. If some different
input signal produce same output signal then by processing output we can not say which input
That is the system produces an all zero sequence for any input sequence. Since every input sequence
[n]} is called the inverse system. In
Causality :
A system is causal if the output at anytime depends only on values of the input at the present time
and in the past.
All memoryless systems are causal. An accumulator system defined by
is also causal. The system defined by
is noncausal.
For real time system where n actually denoted time causality is mportant. Causality is not an
essential constraint in applications where n is not time, for example, image processing. If we are
doing processing on recorded data, then also causality may not be required.
Stability :
There are several definitions for stability. Here we will consider bounded input bonded output (BIBO)
stability. A system is said to be BIBO stable if every bounded input
say that a signal {x[n]} is bounded if
The moving average system
is stable as y[n] is sum of finite numbers and so it is bounded. The accumulator system defined by
is unstable. If we take {x[n]} = {u[n]}, the unit step then y[0] = 1, y[1] = 2, y[2] = 3,
≥ 0 so y[n]grows without bound.
Time invariance :
A system is said to be time invariant if the behavior and characteristics of the system do not change
with time.Thus a system is said to be time invariant if a time delay or time advance in the input signal
leads to identical delay or advance in the output signal. Mathematically if
then
Let us consider the accumulator system
If the input is now {x1[n]} = {x[n-n0]}
A system is causal if the output at anytime depends only on values of the input at the present time
y[n] = f(x[n], x[n-1],...)
All memoryless systems are causal. An accumulator system defined by
also causal. The system defined by
For real time system where n actually denoted time causality is mportant. Causality is not an
essential constraint in applications where n is not time, for example, image processing. If we are
ocessing on recorded data, then also causality may not be required.
There are several definitions for stability. Here we will consider bounded input bonded output (BIBO)
stability. A system is said to be BIBO stable if every bounded input produces a bounded output. We
say that a signal {x[n]} is bounded if
|x[n]| < M < ∞ for all n
is stable as y[n] is sum of finite numbers and so it is bounded. The accumulator system defined by
{x[n]} = {u[n]}, the unit step then y[0] = 1, y[1] = 2, y[2] = 3, are y[n] = n +1, n
A system is said to be time invariant if the behavior and characteristics of the system do not change
time.Thus a system is said to be time invariant if a time delay or time advance in the input signal
leads to identical delay or advance in the output signal. Mathematically if
{y[n]} = T ({x[n]})
{y[n-n0]} = T({x[n-n0]}) for any n0
Let us consider the accumulator system
]} then the corresponding output is
A system is causal if the output at anytime depends only on values of the input at the present time
For real time system where n actually denoted time causality is mportant. Causality is not an
essential constraint in applications where n is not time, for example, image processing. If we are
There are several definitions for stability. Here we will consider bounded input bonded output (BIBO)
produces a bounded output. We
is stable as y[n] is sum of finite numbers and so it is bounded. The accumulator system defined by
are y[n] = n +1, n
A system is said to be time invariant if the behavior and characteristics of the system do not change
time.Thus a system is said to be time invariant if a time delay or time advance in the input signal
The shifted output signal is given by
The two expression look different, but
by l = k - n0 in the first sum then we see that
Hence, {y[n]} = {y[n-n0]} and the system is time
defined by y[n] = nx[n]
if
while
and so the system is not time-invariant. It is time varying. We can also see this by giving a counter
example. Suppose input is
then output is which is definitely not a shifted version version of all zero sequence.
Linearity :
This is an important property of the system. We will see later that if we have system which is linear
and time invariant then it has a very co
important property of superposition: if an input consists of weighted sum of several signals, the nthe
output is also weighted sum of the responses of the system to each of those input signals.
Mathematically let be the response of the system to the input
response of the system to the input. Then the system is linear if:
Additivity: The response to
Homogeneity: The response to
considering only real signals and
signals.
Continuity: Let us consider
that
Let the corresponding output signals be denoted by
We say that system posseses the continuity property if the response of the system to the limiting
input is limit of the responses.
The additivity and continuity properties can be replaced by requiring that system is additive for
countably infinite number of signals i.e. response to
The shifted output signal is given by
The two expression look different, but infact they are equal. Let us change the index of summation
in the first sum then we see that
]} and the system is time-invariant. As a second example consider the system
invariant. It is time varying. We can also see this by giving a counter
then output is all zero sequence. If the input is
which is definitely not a shifted version version of all zero sequence.
This is an important property of the system. We will see later that if we have system which is linear
and time invariant then it has a very compact representation. A linear system possesses the
important property of superposition: if an input consists of weighted sum of several signals, the nthe
output is also weighted sum of the responses of the system to each of those input signals.
be the response of the system to the input and let
response of the system to the input. Then the system is linear if:
is
is , where is any real number if we are
is any complex number if we are considering complex valued
be countably infinite number of signals such
Let the corresponding output signals be denoted by and
We say that system posseses the continuity property if the response of the system to the limiting
is limit of the responses.
The additivity and continuity properties can be replaced by requiring that system is additive for
mber of signals i.e. response to
infact they are equal. Let us change the index of summation
invariant. As a second example consider the system
invariant. It is time varying. We can also see this by giving a counter
input is
which is definitely not a shifted version version of all zero sequence.
This is an important property of the system. We will see later that if we have system which is linear
mpact representation. A linear system possesses the
important property of superposition: if an input consists of weighted sum of several signals, the nthe
output is also weighted sum of the responses of the system to each of those input signals.
and let be the
is any real number if we are
is any complex number if we are considering complex valued
be countably infinite number of signals such
We say that system posseses the continuity property if the response of the system to the limiting
The additivity and continuity properties can be replaced by requiring that system is additive for
Most of the books do not mention the continuity property. They state only finite additivity and
homogeneity. But from finite additivity we can not deduce countable additivity. This distinction
becomes very important in continuous time systems.
A system can be linear without being time invariant and it can be time invariant without being linear.
If a system is linear, an all zero input sequence will produce a all zero output sequence. Let
denote the all zero sequence, then. If
or,
Consider the system defined by
This system is not linear. This can be verified in several ways. If the input is all zero sequence
output is not an all zero sequence. Alth
system is nonlinear. The output of this system can be represented as sum of a linear system and
another signal equal to the zero input response. In this case the linear system is
and the zero-input response is
Such systems correspond to the class of incrementally linear system. System is linear in term of
differnce signal i.e if we define
and the system is linear.
The Convolution Sum:
The representation of discrete time signals in terms of impulses.
The key idea is to express an arbitrary discrete time signal as weighted sum of time shifted impulses.
Consider the product of signal
and
Using these relations we can write
A graphical illustration is shown below
is
Most of the books do not mention the continuity property. They state only finite additivity and
homogeneity. But from finite additivity we can not deduce countable additivity. This distinction
very important in continuous time systems.
A system can be linear without being time invariant and it can be time invariant without being linear.
If a system is linear, an all zero input sequence will produce a all zero output sequence. Let
all zero sequence, then. If then by homogeneity property
This system is not linear. This can be verified in several ways. If the input is all zero sequence
output is not an all zero sequence. Although the defining equation is a linear equation is x and y the
system is nonlinear. The output of this system can be represented as sum of a linear system and
another signal equal to the zero input response. In this case the linear system is
for all n
Such systems correspond to the class of incrementally linear system. System is linear in term of
differnce signal i.e if we define and. Then in terms of
The representation of discrete time signals in terms of impulses.
The key idea is to express an arbitrary discrete time signal as weighted sum of time shifted impulses.
and the impulse sequence. We know that
A graphical illustration is shown below
Most of the books do not mention the continuity property. They state only finite additivity and
homogeneity. But from finite additivity we can not deduce countable additivity. This distinction
A system can be linear without being time invariant and it can be time invariant without being linear.
If a system is linear, an all zero input sequence will produce a all zero output sequence. Let
then by homogeneity property
This system is not linear. This can be verified in several ways. If the input is all zero sequence , the
ough the defining equation is a linear equation is x and y the
system is nonlinear. The output of this system can be represented as sum of a linear system and
Such systems correspond to the class of incrementally linear system. System is linear in term of
and. Then in terms of
The key idea is to express an arbitrary discrete time signal as weighted sum of time shifted impulses.
(4.1)
Fig 4.1
Given an arbitrary sequence we can write it as a linear combination of shifted unit
impulses , where the weights of their combination are x[k], the kth
term of the sequence.
For any given n, in the summation
there is only one term which is non-zero and so we do not have to worry about the convergence.
Consider the unit step sequence {u[n]}. Since , and , it has
representation
The Discrete Time Impulse response of linear Time Invariant System:
We use linearity property of the system to represent its response in terms of its response shifted
impulse sequences. The time invariance further simplifies their representation. Let be the
input signal and be the output sequence, and T( ) represent the linear system
using (4.1)
Now we use the linearity property of the system we get
Note that without countable additivity property the last step is not justified (From finite additivity we
can not get countable additivity). Let us define
i.e. is the response of the system to a delayed unit sample sequence. Then we see
The output signal is linear combination of the signals.
In general the responses need not be related to each other for different values of k.
However, if linear system is also time-invariant, then these responses are related. Let us define
impulse response (unit sample response)
Then
For the LTI system output {y[n]} is given by
(4.2)
This result is know as convolution of sequences and. Thus output signal for an LTI system is
convolution of input signal and the impulse response. This operation is symbolically
represented by
(4.3)
We see that equation (4.2) expresses the response of an LTI system to an arbitrary signal in terms of
the systems response to unit impulse. Thus an LTI system is completely specified by its impulse
response.
The nth
term in the equation (4.2) is given by
(4.4)
This is known as convolution sum. To convolve two sequences, we have to calculate this convolution
sum for all values of n. Since right hand side is sum of infinite series, we assume that this sum is well
defined.
Example:
Consider and shown below
Fig 4.2
Since only and one non zero we have
These one illustrated below
Fig 4.3
Here we have done calculation according to equation (4.2).
To do calculation according to equation (4.4) we first plot - as function of k and
as function of k for some fixed values of n. Then multiply sequence and term by
term to obtain sequence. Than final the sum of the terms of the sequence. This is illustrated below
Fig 4.4
One can see easily that for other value of n is all zero sequence and for these value
of n, output is zero.
Properties of discrete-time linear convolution and system properties
If and are sequences, then the following useful properties of the discrete
time convolution can be shown to be true
1. Commutativity
2. Associativity
`
3. Distributivity over sequence addition
4. The identity sequence
5. Delay operation
6. Multiplication by a constant
Note that these properties are true only if the convolution sum (4.4) exists for every n.
If the input output relation is defined by convolution i.e. if
For a given sequence , then the system is linear and time invariant. This can be verified using
the properties of the convolution listed above. The impulse response of the systems is obviously.
In terms of LTI system, commutative property implies that we can interchange input and impulse
response.
Fig 4.5
The distributive property implies that parallel interconnection of two LTI system is an LTI system with
impulse response as sum of two impulse responses.
Fig 4.6
The associativity property implies that series connection of two LTI system is an LTI system. Where
impulse response is convolution of individual responses. The commutativity property implies that we
can interchange the order of the two system in series.
Fig 4.7
Since an LTI system is completely characterized by its impulse response, we can specify system-
properties in terms of impulse response.
1. Memoryless system: From equation (4.4) we see that an LTI system is memory less if and only
if.
2. Causality for LTI system: The output of a causal system depends only on preset and past-
values of the input. In order for a system to be causal must not depend on for.
From equation (4.4) we see that for this to be true, all of the terms that multiply
values of for must be zero.
put to get
or
Thus impulse response for a causal LTI system must satisfy the condition h[n] = 0 for n
< 0.
If the impulse response satisfies this condition, the system is causal. For a causal system we
can write
or
We say a sequence is causal if , for n < 0.
3. Stability for LTI system: A system is stable if every bounded input produces a bonded output.
Consider input such that for all n.
Taking absolute value
From triangle inequality for complex numbers we get
Using property that
Since each we get
If the impulse response is absolutely summable, that is
(4.5)
then
and is bounded for all n, and hence system is stable. Therefore equation (4.5) is sufficient
condition for system to be stable. This condition is also necessary. This is prove by showing that if
condition (4.5) is violated then we can find a bounded input which produces an unbounded output.
Let
Let
This is a bounded sequence
So y[0] is unbounded. Thus, the stability of a discrete time linear time invariant system is equivalent to
absolute summability of the impulse response.
Causal LTI systems described by difference equations
An important subclass of linear time invariant system is one where the input and output sequences
satisfy constant coefficient linear difference equation
(4.6)
The constants, is input sequence and is output sequence. We can solve equation
(4.6) in a manner analogous to the differential equation solution, but for discrete time we can use a
different approach. Assume that. We can write
(4.7)
In order to find we need previous N values of the output. Thus if we know the input
sequence and a set of initial condition we can find values of.
Example: Consider the difference equation
then
Let us take
This system is not linear for all values of the initial condition. For a linear system all zero input
sequence must produce a all zero output sequence. But if C is different from zero, then output
sequence is not an all system is linear. System is not time invariant in general. Suppose input
is than we have
If we use input as then
It is obvious that second sequence is not a shifted version of the first sequence unless. The system is
linear time invariant if we assume initial rest condition, i.e. if then. With initial rest
condition the system described by constant coefficient-linear difference equation is linear, time
invariant and causal.
The equation of the form (4.7) is called recursive equation if , since it specifies a recursive
algorithm for finding out the output sequence. In special case
Here is completely specified in terms of the input. Thus this equation is called non
equation. If input , then we see that the output is equal to impulse response
The impulse response is non-zero for finitely many values. A system with the property that impulse
response is non-zero only for finitely many values is known as finite impulse response (FIR) system. A
system described by non-recursive equation is always F
generally has a response which is non
infinite impulse response system (IIR). A system described by recessive equation may have a finite
impulse response.
The Discrete Time Fourier Transform
In the previous chapter we used the time domain representation of the signal. Given any signal {x[n]}
we can write it as linear combination of basic signals.
been found very useful is frequency domain representation. In the mid 1960s an algorithm for
calculation of the Fourier transform was discovered, known as the Fast
algorithm. This spurred the development of digital signal processing in many areas.
The Fourier representation of signals derives its importance from the fact that exponential signals
are eigenfunctions for the discrete time LTI systems. What we mean by this is that if
signal to an LTI system then output is given by. Let us co
Then the output is given by
=
=
=
=
where assuming that the summation in right
output is same exponential sequence multiplied by a constant that depends
algorithm for finding out the output sequence. In special case , we have
is completely specified in terms of the input. Thus this equation is called non
, then we see that the output is equal to impulse response
zero for finitely many values. A system with the property that impulse
zero only for finitely many values is known as finite impulse response (FIR) system. A
recursive equation is always FIR. A system described recursive equation
generally has a response which is non-zero for infinite duration and such systems one known as
infinite impulse response system (IIR). A system described by recessive equation may have a finite
The Discrete Time Fourier Transform
In the previous chapter we used the time domain representation of the signal. Given any signal {x[n]}
we can write it as linear combination of basic signals. Another representation of signals that has
useful is frequency domain representation. In the mid 1960s an algorithm for
calculation of the Fourier transform was discovered, known as the Fast-Fourier Transform (FFT)
algorithm. This spurred the development of digital signal processing in many areas.
The Fourier representation of signals derives its importance from the fact that exponential signals
are eigenfunctions for the discrete time LTI systems. What we mean by this is that if
signal to an LTI system then output is given by. Let us consider an LTI system with impulse response.
assuming that the summation in right-hand side converges. Thus
output is same exponential sequence multiplied by a constant that depends on the value of.
Fig 5.1
(4.8)
is completely specified in terms of the input. Thus this equation is called non-recursive
, then we see that the output is equal to impulse response
zero for finitely many values. A system with the property that impulse
zero only for finitely many values is known as finite impulse response (FIR) system. A
IR. A system described recursive equation
zero for infinite duration and such systems one known as
infinite impulse response system (IIR). A system described by recessive equation may have a finite
In the previous chapter we used the time domain representation of the signal. Given any signal {x[n]}
Another representation of signals that has
useful is frequency domain representation. In the mid 1960s an algorithm for
Fourier Transform (FFT)
algorithm. This spurred the development of digital signal processing in many areas.
The Fourier representation of signals derives its importance from the fact that exponential signals
are eigenfunctions for the discrete time LTI systems. What we mean by this is that if is input
nsider an LTI system with impulse response.
hand side converges. Thus
on the value of.
The constant for a specified value of
In the analysis of LTI system, the usefulness of decomposing a more general signal in terms of
eigenfunctions can be seen from the following example. Let
combination of two exponentials
From the eigenfunction property and superposition property the response
More generally if
then
Thus if input signal can be represented by a linear combination of exponential signals, the output can
also be represented by a linear combination of same exponentials, moreover the
linear combination in the output is obtained by multiplying,
representation by corresponding eigen value
represent a large class of signals in terms of comp
representation of aperiodic signals in terms of signals.
The Discrete Time Fourier Transform (DTFT)
Here we take the exponential signals to be
The representation is motivated by the Harmonic analysis, but instead of following the historical
development of the representation we give directly the defining equation.
Let be discrete time signal such that
summable.
The sequence can be represented by a Fourier integral of the form
where
Equation (5.1) and (5.2) give the Fourier representation of the signal. Equation (5.1) is referred as
synthesis equation or the inverse discrete time Fourier transform (IDTFT) and
transform in the analysis equation.
Fourier transform of a signal in general is a complex valued function, we can write
for a specified value of is the eigenvalue associated with eigenfunction.
In the analysis of LTI system, the usefulness of decomposing a more general signal in terms of
can be seen from the following example. Let correspond to a linear
combination of two exponentials
From the eigenfunction property and superposition property the response is given by
=
=
Thus if input signal can be represented by a linear combination of exponential signals, the output can
also be represented by a linear combination of same exponentials, moreover the coefficient of the
linear combination in the output is obtained by multiplying, , the coefficient in the input
representation by corresponding eigen value The procedure outlined above is useful if we can
represent a large class of signals in terms of complex exponentials. In this chapter we will consider
representation of aperiodic signals in terms of signals.
The Discrete Time Fourier Transform (DTFT)
Here we take the exponential signals to be where is a real number.
motivated by the Harmonic analysis, but instead of following the historical
development of the representation we give directly the defining equation.
be discrete time signal such that that is sequence is absolutely
be represented by a Fourier integral of the form
Equation (5.1) and (5.2) give the Fourier representation of the signal. Equation (5.1) is referred as
synthesis equation or the inverse discrete time Fourier transform (IDTFT) and equation (5.2)is Fourier
Fourier transform of a signal in general is a complex valued function, we can write
is the eigenvalue associated with eigenfunction.
In the analysis of LTI system, the usefulness of decomposing a more general signal in terms of
correspond to a linear
is given by
Thus if input signal can be represented by a linear combination of exponential signals, the output can
coefficient of the
, the coefficient in the input
The procedure outlined above is useful if we can
lex exponentials. In this chapter we will consider
motivated by the Harmonic analysis, but instead of following the historical
sequence is absolutely
(5.1)
(5.2)
Equation (5.1) and (5.2) give the Fourier representation of the signal. Equation (5.1) is referred as
equation (5.2)is Fourier
(5.3)
where is the real part of
use a polar form
where is magnitude and
simply, the spectrum to refer to. Thus
called the phase spectrum.
From equation (5.2) we can see that
interpret (5.1) as Fourier coefficients in the representation of a perotic function. In the Fourier series
analysis our attention is on the periodic function, here we are concerned with the representation of
the signal. So the roles of the two equation are interchanged compared to the Fourier series analysis
of periodic signals.
Now we show that if we put equation (5.2) in equation (5.1) we indeed get the signal.
Let
where we have substituted
Since we have used n as index on the left hand side we have used m as the index variable for the sum
defining the Fourier transform. Under our assumption that
we can interchange the order of integration and summation. Thus
The integral with the parentheses can be evaluated as
if then
and
if then
=
=
= 0
Thus in equation (5.5) there is only one non
and is imaginary part of the function. We can also
is the phase of. We also use the term Fourier spectrum or
simply, the spectrum to refer to. Thus is called the magnitude spectrum and
From equation (5.2) we can see that is a periodic function with period i.e.. We can
interpret (5.1) as Fourier coefficients in the representation of a perotic function. In the Fourier series
analysis our attention is on the periodic function, here we are concerned with the representation of
signal. So the roles of the two equation are interchanged compared to the Fourier series analysis
Now we show that if we put equation (5.2) in equation (5.1) we indeed get the signal.
from (5.2) into equation (5.1) and called the result as.
Since we have used n as index on the left hand side we have used m as the index variable for the sum
defining the Fourier transform. Under our assumption that sequence is absolutely summable
he order of integration and summation. Thus
The integral with the parentheses can be evaluated as
Thus in equation (5.5) there is only one non-zero term in RHS, corresponding to
is imaginary part of the function. We can also
(5.4)
is the phase of. We also use the term Fourier spectrum or
is
i.e.. We can
interpret (5.1) as Fourier coefficients in the representation of a perotic function. In the Fourier series
analysis our attention is on the periodic function, here we are concerned with the representation of
signal. So the roles of the two equation are interchanged compared to the Fourier series analysis
equation (5.1) and called the result as.
Since we have used n as index on the left hand side we have used m as the index variable for the sum
sequence is absolutely summable
(5.5)
, and we
get. This result is true for all values of n and so equation (5.1) is indeed a representation of
signal in terms eigenfunctions
In above demonstration we have as
of signals which can be represented by equation (5.1) is equivalent to considering the convergence of
the infinite sum in equation (5.2). If we fix a value of
complex valued series, whose partial sum is given by
The limit as if the partial sum
Since the limit
exists for every. Furthermore it can be shown that the series converges uniformly to a continuous
function of.
If a sequence has only finitely many non
transform exists. Since a stable sequence is by defin
Fourier transform also exits.
Example: Let
Fourier transform of this sequence will exist if it is absolutely summable. We have
This is a geometric series and sum exists if
Thus the Fourier transform of the sequence
get. This result is true for all values of n and so equation (5.1) is indeed a representation of
in terms eigenfunctions
In above demonstration we have assumed that is absolutely summable. Determining the class
of signals which can be represented by equation (5.1) is equivalent to considering the convergence of
the infinite sum in equation (5.2). If we fix a value of then, RHS of equation (5.2) is a
plex valued series, whose partial sum is given by
if the partial sum exists if the series is absolutely summable.
by triangle inequality
exists by our assumption the limit
every. Furthermore it can be shown that the series converges uniformly to a continuous
If a sequence has only finitely many non-zero terms then it is absolutely summable and so the Fourier
transform exists. Since a stable sequence is by definition, an absolutely summable sequence, its
Fourier transform of this sequence will exist if it is absolutely summable. We have
This is a geometric series and sum exists if , in that case
transform of the sequence exists if. The Fourier transform is
get. This result is true for all values of n and so equation (5.1) is indeed a representation of
is absolutely summable. Determining the class
of signals which can be represented by equation (5.1) is equivalent to considering the convergence of
then, RHS of equation (5.2) is a
exists if the series is absolutely summable.
every. Furthermore it can be shown that the series converges uniformly to a continuous
zero terms then it is absolutely summable and so the Fourier
ition, an absolutely summable sequence, its
exists if. The Fourier transform is
Where the last equality follows from sum of a geometric series, which exists if
Absolute summability is a sufficient condition for the existence of a Fourier transform. Fourier
transform also exists for square summable sequence.
For such signals the convergence is not uniform. This has implications in the design of discrete
system for filtering.
We also deal with signals that are neither so absolutely summable nor square summable. To deal
with some of these signals we allow impulse functions, which i
generalized function as a Fourier transform. The impulse function is defined by the following
properties
(a)
(b)
convolution property)
(c) if
Since is a periodic function,
If we substitute this in equation (5.1) we get
equality follows from sum of a geometric series, which exists if
Absolute summability is a sufficient condition for the existence of a Fourier transform. Fourier
transform also exists for square summable sequence.
convergence is not uniform. This has implications in the design of discrete
We also deal with signals that are neither so absolutely summable nor square summable. To deal
with some of these signals we allow impulse functions, which is not an ordinary function but a
generalized function as a Fourier transform. The impulse function is defined by the following
if is continuous at ;(shifting or
if is continuous at
periodic function, let us consider
If we substitute this in equation (5.1) we get
(5.6)
i.e..
Absolute summability is a sufficient condition for the existence of a Fourier transform. Fourier
convergence is not uniform. This has implications in the design of discrete
We also deal with signals that are neither so absolutely summable nor square summable. To deal
s not an ordinary function but a
generalized function as a Fourier transform. The impulse function is defined by the following
;(shifting or
(5.7)
Since there is only one impulse in the interval of integration.
Fourier transform of a signal such that
As a generalization of the above example consider a sequence
substituting this in equation (5.1) we get
as only one term corresponding to
integration
So the signal is when Fourier transform is given by (5.8). More generally if
an arbitrary set if complex exponentials
Thus its Fourier transform is
Thus is a periodic impulse train, with impulses located at
each of the complex exponentials and at all points that are multiples of
An interval of contains exactly one impulse from each of the summation in RHS of (5.9)
Example: Let
Hence
Since there is only one impulse in the interval of integration. Thus we can say that (5.7) represents
that for all.
As a generalization of the above example consider a sequence whose Fourier transform is
substituting this in equation (5.1) we get
will be there in the interval of the
when Fourier transform is given by (5.8). More generally if x[n]
an arbitrary set if complex exponentials
is a periodic impulse train, with impulses located at the frequencies
each of the complex exponentials and at all points that are multiples of from these frequencies.
contains exactly one impulse from each of the summation in RHS of (5.9)
Thus we can say that (5.7) represents
whose Fourier transform is
(5.8)
x[n] is sum of
(5.9)
of
from these frequencies.
contains exactly one impulse from each of the summation in RHS of (5.9)
Properties of the Discrete Time Fourier Transform:
In this section we use the following notation. Let
denoted by and. The notation
is used to say that left hand side is the signal x[n] whose DTFT is
1. Periodicity of the DTFT
As noted earlier that the DTFT
different from the continuous time Fourier transform of a signal.
2. Linearity of the DTFT:
If
and
then
This follows easily from the defining equation (5.2).
3. Conjugation of the signal:
If
then
where * denotes the complex conjugate. We have DTFT of
4. Time Reversal
The DTFT of the time reversal sequence is
the Discrete Time Fourier Transform:
In this section we use the following notation. Let and be two signal, then their DTFT is
and. The notation
is used to say that left hand side is the signal x[n] whose DTFT is is given at right hand
is a periodic function of with period. This property is
different from the continuous time Fourier transform of a signal.
follows easily from the defining equation (5.2).
where * denotes the complex conjugate. We have DTFT of
The DTFT of the time reversal sequence is
be two signal, then their DTFT is
is given at right hand side.
with period. This property is
Let us change the index of summation as
5. Symmetry properties of the Fourier Transform:
If x[n] is real valued than
This follows from property 3. If x[n]
hence
expressing in real and imaginary parts we see that
which implies
and
That is real part of the Fourier transform is an even function of
function of.
The magnitude spectrum is given by
Hence magnitude spectrum of a real signal is an even function of.
The phase spectrum is given by
change the index of summation as
5. Symmetry properties of the Fourier Transform:
This follows from property 3. If x[n] is real valued then , so
in real and imaginary parts we see that
That is real part of the Fourier transform is an even function of and imaginary part is an odd
function of.
The magnitude spectrum is given by
Hence magnitude spectrum of a real signal is an even function of.
and
and imaginary part is an odd
function of.
Hence magnitude spectrum of a real signal is an even function of.
Thus the phase spectrum is an odd function of. We denote the symmetric and antisymmetric part of
a function by
Then using property (2) and (3) we see that
and using property (2) and (4) we can see that
6. Time shifting and frequency shifting:
These can be proved very easily by direct substitution of
and in equation (5.1).
7. Differencing and summation:
This follows directly from
Consider next the signal defined by
since , we are tempted to conclude that the DTFT of
Thus the phase spectrum is an odd function of. We denote the symmetric and antisymmetric part of
Then using property (2) and (3) we see that
property (2) and (4) we can see that
Time shifting and frequency shifting:
These can be proved very easily by direct substitution of in equation(5.2)
in equation (5.1).
This follows directly from linearity property 2.
defined by
, we are tempted to conclude that the DTFT of is DTFT of
Thus the phase spectrum is an odd function of. We denote the symmetric and antisymmetric part of
in equation(5.2)
linearity property 2.
is DTFT of
divided by. This is not entirely true as it ignores the possibility of a dc or average term that can result
from summation. The precise relationship is
We omit the proof of this property.
If we take then we get
8. Time and frequency scaling:
For continuous time signals we know that the Fourier transform of
define a signal we run into
integer say , then we get signal. This consists of taking
Thus the DTFT of this signal looks similar to the Fourier transform of a sampled signal. The result th
resembles the continuous time signal is obtained if we define a signal
For example
divided by. This is not entirely true as it ignores the possibility of a dc or average term that can result
. The precise relationship is
We omit the proof of this property.
then we get
For continuous time signals we know that the Fourier transform of is given by. However if we
we run into difficulty as the index must be an integer. Thus if
, then we get signal. This consists of taking sample of the original signal.
Thus the DTFT of this signal looks similar to the Fourier transform of a sampled signal. The result th
resembles the continuous time signal is obtained if we define a signal by
For example is illustrated below
Fig 5.2
divided by. This is not entirely true as it ignores the possibility of a dc or average term that can result
We omit the proof of this property.
is given by. However if we
must be an integer. Thus if is an
sample of the original signal.
Thus the DTFT of this signal looks similar to the Fourier transform of a sampled signal. The result that
The signal is obtained by inserting
Here we can note the time frequency uncertainly. Since
Fourier transform is compressed.
9. Diffentiation in frequency domain
Differentiating both sides with respect to
multiplying both sides by j we obtain
10. Passeval's relation:
We have
interchanging summation and integration we get
is obtained by inserting zeroes between successive value if signal.
the time frequency uncertainly. Since is expanded sequence, the
Diffentiation in frequency domain
Differentiating both sides with respect to , we obtain
multiplying both sides by j we obtain
interchanging summation and integration we get
zeroes between successive value if signal.
is expanded sequence, the
11. Convolution property:
This is the eigenfunction property of the complex exponential mentioned in the beginning of the
chapter. The fourier syntaxis equation (5.1) for the x[n]
of in terms of linear combinations of complex exponential with amplitude proportional to.
Each of these complex exponential is an eigenfunction of the LTI system and so the
amplitude in the decomposition of
Fourier transform of the impulse response. We prove this formally. The output
terms of convolution sum, so
interchanging order of the summation
Let then
Thus if
then
This is the eigenfunction property of the complex exponential mentioned in the beginning of the
syntaxis equation (5.1) for the x[n] can be interpreted as a representation
in terms of linear combinations of complex exponential with amplitude proportional to.
Each of these complex exponential is an eigenfunction of the LTI system and so the
in the decomposition of will be , where
Fourier transform of the impulse response. We prove this formally. The output
interchanging order of the summation
and we get
This is the eigenfunction property of the complex exponential mentioned in the beginning of the
can be interpreted as a representation
in terms of linear combinations of complex exponential with amplitude proportional to.
Each of these complex exponential is an eigenfunction of the LTI system and so the
is the
is given is
convolution in time domain becomes multiplication in the frequency domain. The fourier transform
of the impulse response is known as frequency response of the system.
12. The Modulation or windowing property
Let us find the DTFT of product of two sequences
Substituting for x[n] in terms of IDFT we get
interchanging order of integration and summation
This looks like convolution of two
and one periodic functions, and equation (5.21) is called periodic convolution. Thus
where denotes periodic convolution.
We summarize these properties in Table (5.1)
Table 5.1: Properties of Discrete time Fourier Transform
Aperiodic signal
convolution in time domain becomes multiplication in the frequency domain. The fourier transform
is known as frequency response of the system.
Modulation or windowing property
Let us find the DTFT of product of two sequences
in terms of IDFT we get
interchanging order of integration and summation
This looks like convolution of two functions, only the interval of integration is
one periodic functions, and equation (5.21) is called periodic convolution. Thus
denotes periodic convolution.
We summarize these properties in Table (5.1)
Discrete time Fourier Transform
Discrete time fourier transform
(5.20)
convolution in time domain becomes multiplication in the frequency domain. The fourier transform
to.
one periodic functions, and equation (5.21) is called periodic convolution. Thus
denotes periodic convolution.
The frequency response of systems characterized by linear constant coefficient difference
equation.
As we have seen earlier, constant coefficient linear difference equation with zero initial condition can
be used to describe some linear time invariant systems.
The input-output and
We assume that Fourier transforms of
response of the system) exist, then convolution property implies that
The frequency response of systems characterized by linear constant coefficient difference
have seen earlier, constant coefficient linear difference equation with zero initial condition can
be used to describe some linear time invariant systems.
are related by
We assume that Fourier transforms of and , ( is the impulse
response of the system) exist, then convolution property implies that
The frequency response of systems characterized by linear constant coefficient difference
have seen earlier, constant coefficient linear difference equation with zero initial condition can
(5.22)
is the impulse
Taking fourier transform of both sides of equation (5.22) and using linearity and time shifting
property of the Fourier transform we get
or
Thus we see that the frequency response is ratio of polynomials in the variable. The numerator
coefficients are the coefficients of
coefficients of in equation (5.22). Thus we can write the frequency resp
inspection.
Example 2: Consider an LTI system initially at rest described by the difference equation
The frequency response of the system is
We can use the inverse fourier transform to get the impulse response
Discrete Fourier series Representation of a periodic signal
Suppose that is a periodic signal with period N, that is
As is continues time periodic signal, we would like to represent
complex exponential signals are given by
All these signals have frequencies is that are multiples of the some fundamental frequency,
Taking fourier transform of both sides of equation (5.22) and using linearity and time shifting
property of the Fourier transform we get
see that the frequency response is ratio of polynomials in the variable. The numerator
in equation (5.22) and denominator coefficients are the
in equation (5.22). Thus we can write the frequency resp
Consider an LTI system initially at rest described by the difference equation
The frequency response of the system is
We can use the inverse fourier transform to get the impulse response
Representation of a periodic signal
is a periodic signal with period N, that is
As is continues time periodic signal, we would like to represent in terms of discrete time
complex exponential signals are given by
All these signals have frequencies is that are multiples of the some fundamental frequency,
Taking fourier transform of both sides of equation (5.22) and using linearity and time shifting
(5.23)
see that the frequency response is ratio of polynomials in the variable. The numerator
in equation (5.22) and denominator coefficients are the
in equation (5.22). Thus we can write the frequency response by
Consider an LTI system initially at rest described by the difference equation
in terms of discrete time
(6.1)
All these signals have frequencies is that are multiples of the some fundamental frequency, , and
thus harmonically related.
These are two important distinction between continuous time and discrete time complex exponential.
The first one is that harmonically related continuous time complex exponential are all
distinct for different values of k , while there are only N different signals in the set.
The reason for this is that discrete time complex exponentials which differ in frequency by integer
multiple of are identical. Thus
So if two values of k differ by multiple of N , they represent the same signal. Another difference
between continuous time and discrete time complex exponential is that for
different k have period which changes with k. In discrete time exponential, if k and N are
relative prime than the period is N and not N/k. Thus if N is a prime number, all the complex
exponentials given by (6.1) will have period N. In a manner analogous to the continuous time, we
represent the periodic signal as
(6.2)
where
(6.3)
In equation (6.2) and (6.3) we can sum over any consecutive N values. The equation (6.2) is synthesis
equation and equation (6.3) is analysis equation. Some people use the faction 1 /N in analysis
equation. From (6.3) we can see easily that
Thus discrete Fourier series coefficients are also periodic with the same period N.
Example 1:
So, and , since the signal is periodic with periodic with period 5, coefficients
are also periodic with period 5, and.
Now we show that substituting equation (6.3) into (6.2) we indeed get.
interchanging the order of summation we get
(6.4)
Now the sum
if n - m multiple of N
and for ( n - m ) not a multiple of N this is a geometric series, so sum is
As m varies from 0 to N - 1, we have only one value of m namely m = n , for which the inner sum if
non-zero. So we set the RHS of (6.4) as.
Properties of Discrete-Time Fourier Series
Here we use the notation similar to last chapter. Let be periodic with period N and discrete
Fourier series coefficients be then the write
where LHS represents the signal and RHS its DFS coefficients
1. Periodicity DFS coefficients:
As we have noted earlier that DFS Coefficients are periodic with period N.
2. Linearity of DFS:
If
If both the signals are periodic with same period N then
3. Shift of a sequence:
(6.5)
(6.6)
To prove the first equation we use equation (6.3). The DFS coefficients are given by
let n - m = l , we get
since is periodic we can use any N consecutive values, then
We can prove the relation (6.6) in a similar manner starting from equation (6.3)
4. Duality:
From equation (6.2) and (6.3) we can see that synthesis and analysis equation differ only in sign of the
exponential and factor 1/N. If is periodic with period N , then is also periodic with
period N. So we can find the discrete fourier series coefficients of sequence.
From equation (6.2) we see that
Thus
Interchanging the role of k and n we get
comparing this with (6.3) we see that DFS coefficients of are the original
periodic sequence is reversed in time and multiplied by N. This is known as duality property. If
(6.7)
then
(6.8)
5. Complex conjugation of the periodic sequence:
substituting in equation (6.3) we get
6. Time reversal:
From equation (6.3) we have the DFS coefficient
putting m = - n we get
Since is periodic, we can use any N consecutive values
7. Symmetry properties of DFS coefficient:
In the last chapter we discussed some symmetry properties of the discrete time Fourier transform of
aperiodic sequence. The same symmetry properties also hold for DFS coefficients and their derivation
is also similar in style using linearity, conjugation and time reversal properties DFS coefficients.
8. Time scaling:
Let us define
sequence is obtained by inserting ( m - 1) zeros between two consecutive values of. Thus
Thus is also periodic, but period is mN. The DFS coefficients are given by
putting
as non zero terms occur only when r = 0
If we define then is periodic with period equal to least common multiple (LCM)
of M and N. The relationship between DFS coefficients is not simple and we omit it here.
9. Difference
This follows from linearity property.
10. Accumulation
Let us define
will be bounded and periodic only if the sum of terms of over one period is zero,
i.e. , which is equivalent to. Assuming this to be true
11. Periodic convolution
Let and be two periodic signals having same period N with discrete Fourier series
coefficients denoted by and respectively. If we form the
product then we want to find out the sequence whose DFS
coefficients are. From the synthesis equation we have
substituting for in terms of we get
interchanging order of summations we get
(6.15)
as inner sum can be recognized as from the synthesis equation. Thus
The sum in the equation (6.15) looks like convolution sum, except that the summation is over one
period. This is known as periodic convolution. The resulting sequence is also periodic with
period N. This can be seen from equation (6.15) by putting m + N instead of m.
The Duality theorem gives analogous result when we multiply two periodic sequences.
The DFS coefficients are obtained by doing periodic convolution of and and
multiplying the result by 1/N. We can also prove this result directly by starting from the analysis
equation. The periodic convolution has properties similar to the aperiodic (linear convolution).It is
cumulative, associative and distributes over additions of two signals.
The properties of DFS representation of periodic sequence are summarized in the Table 6.1
Periodic sequence (period N) DFS coefficients (Period N)
1.
period N
2.
3.
4.
5.
6.
7.
8.
(periodic with period mN)
(viewed as periodic with period mN)
9.
10.
(periodic only if )
11.
12.
13.
14.
15.
16.
17. If is real then
Table 6.1
Fourier Transform of periodic signals
If is periodic with period N, then we can write
Using equation (5.9) we see that
as is periodic with period N.
Example:
Consider the periodic impulse train
then
as only one term corresponding to n = 0 is non zero. Thus the DTFT is
Fourier Representation of Finite Duration sequence
The Discrete Fourier Transform (DFT)
We now consider the sequence such that and. Thus can be take non-
zero values only for. Such sequences are known as finite length sequences, and N is called the length
of the sequence. If a sequence has length M, we consider it to be a length N sequence where. In these
cases last ( N - M ) sample values are zero. To each finite length sequence of length N we can always
associate a periodic sequence defined by
(6.16)
Note that defined by equation (6.16) will always be a periodic sequence with period N,
whether is of finite length N or not. But when has finite length N, we can recover the
sequence from by defining
(6.17)
This is because of has finite length N , then there is no overlap between terms
and for different values of.
Recall that if
n = kN + r, where
then n modulo N = r ,
i.e. we add or subtract multiple of N from n until we get a number lying between 0 to N - 1. We will
use ((n))N to denote n modulo N. Then for finite length sequences of length N equation (6.16) can be
written as
(6.18)
We can extract from using equation (6.17). Thus there is one-to- one correspondance
between finite length sequences of length N , and periodic sequences of period N.
Given a finite length sequence we can associate a periodic sequence with it.
This periodic sequence has discrete Fourier series coefficients which are also periodic with
period N.From equations (6.2) and (6.3) we see that we need values of for
and for 0 = k = N - 1. Thus we define discrete Fourier transform of finite length
sequence as
where is DFS coefficient of associated periodic sequence. From we can get
by the relation.
then from this we can get using synthesis equation (6.2) and finally using equation
(6.17). In equations (6.2) and (6.3) summation interval is 0 to N - 1, we can write X [k ] directly in terms
of x[n], and x[n] directly in terms of X[k] as
For convenience of notation, we use the complex quantity
(6.19)
with this notation, DFT analysis and synthesis equations are written a follows
Analysis equation:
(6.20)
Synthesis equation:
(6.21)
If we use values of k and n outside the interval 0 to N - 1 in equation (6.20) and (6.21), then we will
not get values zero, but we will get periodic repetition of and respectively. In defining
DFT, we are concerned with values only in interval 0 to N - 1. Since a sequence of length M can also be
considered a sequence of length , we also specify the length of the sequence by saying N-
point-DFT, of sequence.
Sampling of the Fourier transform:
For sequence of length N, we have two kinds of representations, namely, discrete time
Fourier transform and discrete Fourier transform. The DFT values can be considered
as samples of
(as x[n] = 0 n < 0, for n < 0, and n > N - 1)
(6.22)
Thus is is obtained by sampling at.
Properties of the discrete Fourier transform
Since discrete Fourier transform is similar to the discrete Fourier series representation, the properties
are similar to DFS representation. We use the notation
to say that are DFT coefficient of finite length sequence.
1. Linearity
If two finite length sequence have length M and N , we can consider both of them with length greater
than or equal to maximum of M and N. Thus if
then
where all the DFTs are N-point DFT. This property follows directly from the equation (6.20)
2. Circular shift of a sequence
If we shift a finite length sequence of length N , we face some difficulties. When we shift it in
right direction the length of the sequence will becam according to
definition. Similarly if we shift it left , if may no longer be a finite length sequence
as may not be zero for n < 0. Since DFT coefficients are same as DFS coefficients, we define
a shift operation which looks like a shift of periodic sequence. From we get the periodic
sequence defined by
We can shift this sequence by m to get
Now we retain the first N values of this sequence
This operation is shown in figure below for m = 2, N = 5.
Fig 6.1
We can see that is not a shift of sequence. Using the propertiesof the modulo arithmetic we
have
and
(6.23)
The shift defined in equation (6.23) is known as circular shift. This is similar to a shift of sequence in a
circular register.
Fig 6.2
3. Shift property of DFT
From the definition of the circular shift, it is clear that it corresponds to linear shift of the associated
periodic sequence and so the shift property of the DFS coefficient will hold for the circular shift. Hence
(6.24)
and
(6.25)
4. Duality
We have the duality for the DFS coefficient given by , retaining one period of
the sequences the duality property for the DFT coefficient will become
5. Symmetry properties
We can infer all the symmetry properties of the DFT from the symmetry properties of the associated
periodic sequence and retaining the first period. Thus we have
and
We define conjugate symmetric and anti-symmetric points in the first period 0 to N - 1 by
Since
the above equation similar to
(6.26)
(6.27)
and are referred to as periodic conjugate symmetric and periodic conjugate anti-
symmetric parts of. In terms if these sequence the symmetric properties are
6. Circular convolution
We saw that multiplication of DFS coefficients corresponds of periodic convolution of the sequence.
Since DFT coefficients are DFS coefficients in the interval, , they will correspond to DFT of
the sequence retained by periodically convolving associated periodic sequences and retaining their first
period.
Periodic convolution is given by
using properties of the modulo arithmetic
and then
Since we get
The convolution defined by equation (6.28) is known as N-point-circular convolution of
sequence and , where both the sequence are considered sequence of length N. From
the periodic convolution property of DFS it is clear that DFT of is. If we use the
notation to denote the N point circular convolution we see that
(6.29)
In view of the duality property of the DFT we have
(6.30)
Properties of the Discrete Fourier transform are summarized in the table 6.2
Finite length sequence (length N) N-point DFT (length N)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14. If is real sequence
Linear convolution using the Discrete Fourier Transform
Output of a linear time invariant-system is obtained by linear convolution of input signal with the
impulse response of the system. If we multiply DFT coefficients, and then take inverse transform we
will get circular convolution. From the examples it is clear that result of circular convolution is
different from the result of linear convolution of two sequences. But if we modify the two sequence
appropriately we can get the result of circular convolution to be same as linear convolution. Our
interest in doing linear convolution results form the fact that fast algorithms for computing DFT and
IDFT are available. These algorithms will be discussed in a later chapter. Here we show how we can
make result of circular convolution same as that of linear convolution.
If we have sequence of length L and a sequence of length M , the sequence
obtained by linear convolution has length ( L + M - 1).
This can be seen from the definition
(6.31)
as x[k] = 0 for. For hence. Similarly
for , so. Hence is possibly nonzero only for.
Now consider a sequence , DTFT is given by
writing
We get
If we take
we see that
Comparing this with the DFT equation (6.), we see that
can be seen as DFT coefficients of a sequence
(6.32)
obviously if has length less then or equal to N , then
However, if the length of
of l.
The sequence in equation (6.31) has the discrete Fourier transform
The N-point DFT of sequence is
where and are N
resulting as the inverse DFT of
From the circular convolution property of the DFT we have
Thus, the circular convolution of two
followed time aliasing, defined by equation (6.32). If N is greater than or equal to (
there will be no time aliasing as the linear convolu
we can use circular convolution for linear convolution by padding sufficient number of zeros at the
end of a finite length sequence. We can use DFT algorithm for calculating the circular convolution.
Definition of the Z-transform
We saw earlier that complex exponential of the from
We can generalize this for signals of the form
=
=
is greater than may not be equal to
in equation (6.31) has the discrete Fourier transform
sequence is
are N-point DFTs of and respectively. The sequence
is then by equation (6.32).
From the circular convolution property of the DFT we have
Thus, the circular convolution of two-finite length sequences can be viewed as linear convolution,
followed time aliasing, defined by equation (6.32). If N is greater than or equal to ( L +
there will be no time aliasing as the linear convolution produces a sequence of length (
we can use circular convolution for linear convolution by padding sufficient number of zeros at the
end of a finite length sequence. We can use DFT algorithm for calculating the circular convolution.
We saw earlier that complex exponential of the from is an eigen function of for a LTI System.
We can generalize this for signals of the form where, is a complex number.
for all values
respectively. The sequence
finite length sequences can be viewed as linear convolution,
M - l ), then
tion produces a sequence of length ( L + M - l ). Thus
we can use circular convolution for linear convolution by padding sufficient number of zeros at the
end of a finite length sequence. We can use DFT algorithm for calculating the circular convolution.
is an eigen function of for a LTI System.
=
=
where
Thus if the input signal is then output signal is. For
(7.1) is same as the discrete-time fourier transform. The
bilateral z-transform of the sequence. We define for any
where is a complex variable. Writing
and is angle of.
=
=
This shows that is Fourier transform of the sequence. When
reduces to the Fourier transform of. From equation (7.3) we see that for convergence of z
that Fourier transform of the sequence
others. The values of z - for which
the ROC contains unit circle (i.e.
converges. Following examples show that we must specify ROC to completely specify the z
Example 1: Let
=
=
This is a geometric series and converges if
then output signal is. For real (i.e for
time fourier transform. The in equation (7.1) is known as the
transform of the sequence. We define for any sequence of a sequence
is a complex variable. Writing in polar form we get , where
is Fourier transform of the sequence. When the z
to the Fourier transform of. From equation (7.3) we see that for convergence of z
that Fourier transform of the sequence converges. This will happen for some r and not for
is called the region of convergenc
the ROC contains unit circle (i.e. or equivalently then the Fourier transform also
converges. Following examples show that we must specify ROC to completely specify the z
, then
geometric series and converges if or. Then
(7.1)
), equation
in equation (7.1) is known as the
as
(7.2)
is magnitude
(7.3)
the z-transform
to the Fourier transform of. From equation (7.3) we see that for convergence of z-transform
converges. This will happen for some r and not for
is called the region of convergence(ROC). If
then the Fourier transform also
converges. Following examples show that we must specify ROC to completely specify the z-transform.
We see that at , and
zero of and value of where
of a region in Z-plane which lies outside the circle centered at origin and passing through the pole.
Example 2: Let,
=
=
This is a geometric series which converges when
, and at. Values of where is zero is called
where is zero is called a pole of. Here we see that ROC consists
outside the circle centered at origin and passing through the pole.
Fig 7.1
, then
This is a geometric series which converges when , that is Then
(7.4)
is zero is called
is zero is called a pole of. Here we see that ROC consists
outside the circle centered at origin and passing through the pole.
(7.5)
Here the ROC is inside the circle of
form of and are same, but ROC are different and they correspond to two different
sequences. Thus in specifying z-transform, we have to give functional form
convergence.
Now we state some properties of the region of convergence
Properties of the ROC
1. The ROC of consists of an annular region in the z
property follows from equation (7.3), where we see that convergence
2. The ROC does not certain any poles. Since at poles
3. The ROC is a connected region in z
4. If is a right sided sequence, i.e.
the ROC, then all finite values of
For a right sided sequence
If is negative then we can write
Let , with
Fig 7.2
Here the ROC is inside the circle of radius. Comparing equation (7.4) and (7.5) we see that algebraic
are same, but ROC are different and they correspond to two different
transform, we have to give functional form and the region of
Now we state some properties of the region of convergence
consists of an annular region in the z-plane, centered about the origin. This
property follows from equation (7.3), where we see that convergence depends on
The ROC does not certain any poles. Since at poles does not converge.
The ROC is a connected region in z-plane. This property is proved in complex analysis.
is a right sided sequence, i.e. , for , and if the circle
ROC, then all finite values of , for which will also be in the ROC.
is negative then we can write
, with , then, exists if
radius. Comparing equation (7.4) and (7.5) we see that algebraic
are same, but ROC are different and they correspond to two different
and the region of
plane, centered about the origin. This
depends on only.
plane. This property is proved in complex analysis.
, and if the circle is in
will also be in the ROC.
exists if
The first summation is finite as it consists of a finite number of terms. In the second
summation note that each term is less than
assumption that circle with radius
of z such that lies in ROC, except when. At
infinite. So if , i.e. the sequence
5. If is left sided sequence, i.e.
values of function
The proof is similar to the property 4. The point
purely
anticausal
6. If is non zero for,
and/or. In this case the
each term infinite which is the case when
if , and lies in the ROC if.
7. If is two-sided sequence and if circle
annular region in z-plane, which includes. We can express a two sided sequence as sum of a
right sided sequence and a left sided sequence. Then using property 4 and 5 we get this
property. Using
property 2 and 3 we see what ROC will be banded by circles passing
The inverse z-transform
The inverse z-transform is given by
the symbol indicates contour integration, over a counter clockwise contour in the ROC of. If
ratio of polynomials one can use Cauchy integral
alternative procedures also, which will be considered after discussing the properties of z
Properties of the z-transform
We use the notation
to denote z-transform of the sequence.
1. Linearity
is finite.
first summation is finite as it consists of a finite number of terms. In the second
summation note that each term is less than as. Since
assumption that circle with radius lies in ROC, the second sum is also finite. Hence values
lies in ROC, except when. At , the first summation will became
, i.e. the sequence is causal, the value will lie in the ROC.
is left sided sequence, i.e. and lies in the ROC, the
function also lie in the ROC.
The proof is similar to the property 4. The point , will lie in the ROC if the sequence is
, then ROC is entire z-plane except possibly
consists of finite number of terms and therefore it converges if
each term infinite which is the case when is different from 0 or.
lies in the ROC if.
sided sequence and if circle is in ROC, then ROC will consist of
plane, which includes. We can express a two sided sequence as sum of a
right sided sequence and a left sided sequence. Then using property 4 and 5 we get this
property. Using
property 2 and 3 we see what ROC will be banded by circles passing through the poles.
indicates contour integration, over a counter clockwise contour in the ROC of. If
ratio of polynomials one can use Cauchy integral theorem to calculate the contour integral. There are some other
alternative procedures also, which will be considered after discussing the properties of z
transform of the sequence.
is finite.
first summation is finite as it consists of a finite number of terms. In the second
is finite by our
lies in ROC, the second sum is also finite. Hence values
, the first summation will became
will lie in the ROC.
lies in the ROC, the
the ROC.
, will lie in the ROC if the sequence is
plane except possibly ,
of finite number of terms and therefore it converges if
lies in ROC,
is in ROC, then ROC will consist of
plane, which includes. We can express a two sided sequence as sum of a
right sided sequence and a left sided sequence. Then using property 4 and 5 we get this
property. Using
through the poles.
indicates contour integration, over a counter clockwise contour in the ROC of. If
theorem to calculate the contour integral. There are some other
alternative procedures also, which will be considered after discussing the properties of z-transform.
The z-transform of a linear combination of two sequence is given by
The algebraic form follows directly from the definition, equation (7.2). The linear combination is
such that some zero's can cancel the poles, then the region of convergence may be larger. For
example if the linear combination
entire z-plane except at , like individual ROCs are. If the intersection of
set, the z-transform of the linear combination will not exist.
2. Time shifting
If we shift the time sequence, we get
,
and/or
We have
changing variable,
=
=
=
The factor can affect the poles and zeros at
3. Multiplication by a exponential sequence
This follows directly from defining equation (7.2).
4. Differentiation of :
If we differentiate term by term we get
transform of a linear combination of two sequence is given by
ROC contains
The algebraic form follows directly from the definition, equation (7.2). The linear combination is
's can cancel the poles, then the region of convergence may be larger. For
is a finite-length sequence, the ROC is
, like individual ROCs are. If the intersection of and
ransform of the linear combination will not exist.
If we shift the time sequence, we get
except for possible addition or deletion of
the poles and zeros at ,
Multiplication by a exponential sequence
This follows directly from defining equation (7.2).
term by term we get
The algebraic form follows directly from the definition, equation (7.2). The linear combination is
's can cancel the poles, then the region of convergence may be larger. For
length sequence, the ROC is
and is null
If we shift the time sequence, we get
except for possible addition or deletion of
Thus
The ROC does not change (except
analytic function.
5. Conjugation of a complex sequence
=
=
=
Since ROC depends only an magnitude
6. Time Reversal
We have
putting
, except possibly
, ). This follows from the property that
5. Conjugation of a complex sequence
=
=
=
Since ROC depends only an magnitude it does not change.
). This follows from the property that is an
=
=
If we combine it with the previous property, we get
7. Convolution of sequence
The z-transform of the convolution is
Interchanging the order of summation
using time shifting property (or changing index of summation)
=
=
If there is pole-zero cancelation, the ROC will be larger than the common ROC of two sequence.
Convolution property plays an important role in analysis
produces a delay of , has the transfer function
depicted by
If we combine it with the previous property, we get
ROC contains
transform of the convolution is
of summation
using time shifting property (or changing index of summation)
zero cancelation, the ROC will be larger than the common ROC of two sequence.
Convolution property plays an important role in analysis of LTI system. An LTI system, which
, has the transfer function , therefore delay of
zero cancelation, the ROC will be larger than the common ROC of two sequence.
of LTI system. An LTI system, which
units is often
8. Complex convolution theorem
If we multiply two sequences then
This can be proved using inverse z-transform definition.
9. Initial value Theorem
If is zero for , i.e.
Taking limit term by term in , we get the above result.
10. Parseval's relation
These properties are summarized in table 7.1
Table
Methods of inverse z-transform
We can use the contour integration and the equation (7.6) to calculate inverse z
equation has to be evaluated for all values of
Here we give two simple methods for the inverse transform computation.
1. Inverse transform by partial fraction expansion
This is method is useful when z-transform is ratio of polynomials. A rational
Fig 7.3
ROC contains
transform definition.
is causal, then
, we get the above result.
These properties are summarized in table 7.1
Table 7.1 z-transform properties
We can use the contour integration and the equation (7.6) to calculate inverse z-transform. This
equation has to be evaluated for all values of , which can be quite complicated in many cases.
Here we give two simple methods for the inverse transform computation.
1. Inverse transform by partial fraction expansion
transform is ratio of polynomials. A rational can be expressed
transform. This
, which can be quite complicated in many cases.
can be expressed
as
where and are polynomials in. If degree
greater than or equal to the degree N of the denominator polynomial
by and re-express as
where the degree of polynomial
all poles are simple. Then
where
Example: Let
The partial fraction expression is
The inverse z-transform depends on the ROC. If ROC is
term is outside a circle(so that common ROC is outside a circle), sequences are causal. Using linearity
polynomials in. If degree of the numerator polynomial
greater than or equal to the degree N of the denominator polynomial , we can divide
is strictly less than that of. For simplicity let us assume that
=
=
=
=
transform depends on the ROC. If ROC is , then ROCs associated with each
term is outside a circle(so that common ROC is outside a circle), sequences are causal. Using linearity
of the numerator polynomial is
, we can divide
t us assume that
, then ROCs associated with each
term is outside a circle(so that common ROC is outside a circle), sequences are causal. Using linearity
property and z-transform of
If the ROC is , the ROC of the term
and ROC for should be. Hence we get
Similarly if ROC is we get a noncausal sequence
If has multiple poles, the partial fraction has slightly different form. If
order s at , and all other poles are simple Then
where and are obtained as before, the coefficients
If there are more multiple poles, there will be more terms like the third term.
Using linearity and differentiation properties we get some useful z
Sequence
1.
2.
3.
we get
, the ROC of the term should be outside the circle
should be. Hence we get the sequence as
we get a noncausal sequence
has multiple poles, the partial fraction has slightly different form. If
, and all other poles are simple Then
are obtained as before, the coefficients are given by
If there are more multiple poles, there will be more terms like the third term.
Using linearity and differentiation properties we get some useful z-transform pairs given in Table 7.2
Transform ROC
All
All , except 0(if ) or
should be outside the circle ,
has a pole of
If there are more multiple poles, there will be more terms like the third term.
transform pairs given in Table 7.2
if
4.
5.
6.
7.
8.
9.
Table 7.2
2. Inverse Transform via long division
For causal sequence the z-transform
expansion, the coefficient multiplying the term
terms of poles of.
Example 1: Let
This is a causal sequence, long division gives
Table 7.2 Some useful z-transform pairs
Inverse Transform via long division
transform can be exported into a pure series in. In the series
expansion, the coefficient multiplying the term is. If is anticausal then we expand in
This is a causal sequence, long division gives
can be exported into a pure series in. In the series
is anticausal then we expand in
This is a causal sequence, long division gives
This gi
We can see that it is not easy to write the
Example 2:
Using the pure series expansion for
=
=
Analysis of LTI system using z-transform
From the convolution property we have
where are are
and impulse response respectively. The
function of the system. For on the unit circle
response of the system, provided th
A causal LTI system has impulse response
in z-plane including. Thus a discrete time LTI system is causal if and only if ROC is exterior of a circle
which includes infinit
An LTI system is stable if and only if impulse response
equivalent to saying that unit circle is in the ROC of.
For a causal and stable system ROC is outside a circle and ROC contains the unit circle. That means all
the poles are inside the unit circle. Thus a causal LTI system is stable if and if only if all the poles
inside unit circle.
LTI systems characterized by Linear constant coefficient difference equation
For the system characterized by
We take the z-transform of both sides and use linearity and the time shift property to get
This gives
We can see that it is not easy to write the
Using the pure series expansion for with , we obtain
transform
From the convolution property we have
z-transforms of input sequence , output sequence
respectively. The is referred to as system function or transfer
on the unit circle , reduces to the frequency
response of the system, provided that unit circle is in the ROC for.
A causal LTI system has impulse response such that. Thus ROC of is exterior of a circle
plane including. Thus a discrete time LTI system is causal if and only if ROC is exterior of a circle
which includes infinit
An LTI system is stable if and only if impulse response is absolutely summable. This is
equivalent to saying that unit circle is in the ROC of.
For a causal and stable system ROC is outside a circle and ROC contains the unit circle. That means all
he poles are inside the unit circle. Thus a causal LTI system is stable if and if only if all the poles
LTI systems characterized by Linear constant coefficient difference equation
transform of both sides and use linearity and the time shift property to get
,.....
We can see that it is not easy to write the term.
, output sequence
is referred to as system function or transfer
reduces to the frequency
at unit circle is in the ROC for.
is exterior of a circle
plane including. Thus a discrete time LTI system is causal if and only if ROC is exterior of a circle
which includes infinity.
is absolutely summable. This is
equivalent to saying that unit circle is in the ROC of.
For a causal and stable system ROC is outside a circle and ROC contains the unit circle. That means all
he poles are inside the unit circle. Thus a causal LTI system is stable if and if only if all the poles
transform of both sides and use linearity and the time shift property to get
Thus the system function is always a rational function. We can write it by inspection. Numerator
polynomial coefficients are the coefficients of
of. The difference equation by itself does not provide information about the ROC, it can be determined
by conditions like causality and stability.
System Function and block diagram representation
The use of z-transform allows us to replace time domain operation such as convolution time shifting
etc with algebraic operations.
Consider the parallel interconnection if two system, as shown in figure 7.4.
The impulse response of the over all
From linearity of the z-transform,
Similarly, the impulse response of the series connection in figure 7.5 is
=
=
Thus the system function is always a rational function. We can write it by inspection. Numerator
polynomial coefficients are the coefficients of and denominator coefficients are coefficients
of. The difference equation by itself does not provide information about the ROC, it can be determined
by conditions like causality and stability.
System Function and block diagram representation
transform allows us to replace time domain operation such as convolution time shifting
etc with algebraic operations.
Consider the parallel interconnection if two system, as shown in figure 7.4.
Fig 7.4
The impulse response of the over all system is
Similarly, the impulse response of the series connection in figure 7.5 is
Thus the system function is always a rational function. We can write it by inspection. Numerator
and denominator coefficients are coefficients
of. The difference equation by itself does not provide information about the ROC, it can be determined
transform allows us to replace time domain operation such as convolution time shifting
etc with algebraic operations.
From the convolution property.
The z-transform of the interconnection of linear system can be obtained by algebraic
example consider the feed back connection in figure 7.6
We have
or
Even though this course is primarily about the discrete time signal processing, most signals we
encounter in daily life are continuous in time such as speech, music and images. Increasingly discrete
time signals processing algorithms are being used to process such signals. For processing by digital
systems, the discrete time signals are represented in digital form with
binary word. Therefore we need the analog to digital and digital to analog interface circuits to convert
the continuous time signals into discrete time digital form and vice versa. As a result it is necessary to
develop the relations between continuous time and discrete time representations.
Fig 7.5
transform of the interconnection of linear system can be obtained by algebraic
example consider the feed back connection in figure 7.6
Fig 7.6
=
=
=
=
=
Even though this course is primarily about the discrete time signal processing, most signals we
life are continuous in time such as speech, music and images. Increasingly discrete
time signals processing algorithms are being used to process such signals. For processing by digital
systems, the discrete time signals are represented in digital form with each discrete time sample as
binary word. Therefore we need the analog to digital and digital to analog interface circuits to convert
the continuous time signals into discrete time digital form and vice versa. As a result it is necessary to
lations between continuous time and discrete time representations.
transform of the interconnection of linear system can be obtained by algebraic means. For
Even though this course is primarily about the discrete time signal processing, most signals we
life are continuous in time such as speech, music and images. Increasingly discrete-
time signals processing algorithms are being used to process such signals. For processing by digital
each discrete time sample as
binary word. Therefore we need the analog to digital and digital to analog interface circuits to convert
the continuous time signals into discrete time digital form and vice versa. As a result it is necessary to
Sampling of continuous time signals
Let be a continuous time signal that is sampled uniformly at t = nT generating the
sequence where
T is called sampling period, the reciprocal of T is called the sampling frequency. The frequency domain
representation of is given by its Fourier transform
where the frequency-domain representation of is given by its discrete time fourier transform
To establish relationship between the two representation, we use impulse train sampling. This should
be understood as mathematically convenient method for understanding sampling. Actual circuits can
not produce continuous time impulses. A periodic impulse train is given by
(8.1)
Fig 8.1
(8.2)
using sampling property of the impulse , we get
(8.3)
Fig 8.2
From multiplication property, we know that
The Fourier transform of a impulse train is given by
where
Using the property that it follows that
(8.4)
Thus is a periodic function of with period , consisting of superposition of shifted
replicas of scaled by. Figure 8.3 illustrates this for two cases.
Fig 8.3
If or equivalently there is no overlap between shifted replicas
of , where as with , there is overlap. Thus if is faithfully
replicated in and can be recovered from by means of lowpass filtering with gain T
and cut off frequency between and. This result is known as Nyquist sampling theorem.
Sampling Theorem
Let be a bandlimited signal with , for. Then is uniquely determined by its
samples , if
The frequency is called Nyquist rate, while the frequency is called the Nyquist frequency.
The signal can be reconstructed by passing through a lowpass filter.
Fig 8.4
The impulse response of this filter is
(8.5)
Assuming we get
(8.6)
The above expression (8.5) shows that reconstructed continuous time signal is obtained by
shifting in time the impulse response of low pass filter by an amount nT and scaling it in
amplitude by a factor for all integer values n. The interpolation using the impulse
response of an ideal low pass filter in (8.6) is referred to as bandlimited interpolation, since it
implements reconstruction if is bandlimited and sampling frequency satisfies the condition of
the sampling theorem. The reconstruction is in the mean square sense i.e.
The effect of underselling: Aliasing
We have seen earlier that spectrum is not faithfully copied when. The terms in (8.4)
overlap. The signal is no longer recoverable from. This effect, in which individual terms in
equation (8.4) overlap is called aliasing.
For the ideal low pass signal
Hence
Thus at the sampling instants the signal values of the original and reconstructed signal are same for
any sampling frequency.
DTFT of the discrete time signal
Taking continuous time Fourier transform of equation (8.3) we get
(8.7)
Since , we get the DTFT
(8.8)
comparing them we see that
using equation (8.4) we get
or
(8.9)
Comparing equation (8.4) and(8.9) we see that is simply a frequency scaled version
of with frequency scaling specified by. This can be thought of as a normalization of
frequency axis so that frequency in is normalized to in. For the example
in figure 8.3 the is shown in figure (8.5)
From equation (8.5) we see that
Fig 8.5
(8.10)
We refer to the system that implements as ideal continuous-to-discrete time (C/D)
convertor and is depicted in figure (8.6)
Fig 8.6
The ideal system that takes sequence as input and produces given equation (8.5) is
called ideal discrete to continuous time convertor and is depicted in Figure (8.7)
Fig 8.7
Discrete time processing of continuous time signal
Figure (8.8) shows a system for discrete time processing of continuous time system
Fig 8.8
The over all system has as input and as output. We have the following relations among
the signals.
and
If the discrete time system is LTI then we have
combining these equations we get
(8.11)
If , for and we use ideal lowpass reconstruction filter then only the term
for k = 0 is passed by the filter and we get
Thus if is bandlimited and sampling rate is above the Nyquist rate, the output is related to
the input by
where
(8.12)
That is overall system is equivalent to a linear time invariant system for bandlimited signal.
The LTI property of the system depends on two factors. First the discrete time system is LTI and
second the input signals are bandlimited to half the sampling frequency
Example
Let us consider the system in figure 8.8 with
The frequency response is periodic with period. For a bandlimited input signal, sampled above the
Nyquist rate, the overall system will behave like a LTI continuous time system with
Thus the equivalent system is ideal lowpass system with cut off frequency. With a fixed discrete time
filter by changing T we can change the cut off frequency of the equivalent system. Spectra for various
signals are depicted in figure 8.9.
FIGURE 8.9
From figure (8.9) we can see that ever if there is same aliasing due to sampling, if the components are
filtered out by the discrete time system, the over all transfer function will remain same. Thus the
requirement is
instead of for no aliasing.
Continuous time processing of discrete time signals
Consider the system shown in figure (8.10)
Figure 8.10
We have
Therefore the overall system behaves as a discrete time system where frequency response is
(8.13)
Example
Let us consider a discrete time system with frequency response
when is an integer, this system is delay by
but when is not an integer, we can not write the above equation. Suppose that we implement this
using system in figure (8.10). Then we have
(8.14)
So that overall system has frequency response. The equation (8.13) represents a time delay secs
in continuous time whether is integer or not, thus
The signal is bandlimited interpolation of and is obtained by sampling. Thus y[n]
are samples of band limited signal delayed by.
For are depicted in figure (8.11)
Fig 8.11
Sampling of discrete time Signals
In analogy with continuous time sampling, the sampling of a discrete time signal can be represented
as shown in figure 8.12
FIGURE 8.12
(8.15)
In frequency domain, we get
The Fourier transform of sequence is
If or equivalently or there will be no aliasing (i.e non zero
portions of do not overlap) and the signal can be recovered from by passing
through an ideal low-pass filter with gain equal to N and cut off equal to
FIGURE 8.14
If , there will be aliasing, and so will be different from. However as in continuous
time case
independently of whether there is aliasing or not.
For ideal low pass filter
with we get
Discrete time decimation and interpolation
The sampled signal in equation (8.13) has ( N - 1) samples out of every N samples as zeros. We define
a new sequence which retains only the non zero values
(8.17)
this is called a decimated sequence, whatever may be the value of. The DTFT of the decimated
request is given by
since only for multiples of has non zero value,
(8.18)
For the signal shown in figure (8.13) the sequence and its spectrum are illustrated in figure
(8.15)
Fig 8.15
If the original signal was obtained by sampling a continuous time signal, the process of
decimation can be viewed as reduction in the sampling rate by a factor of N. With this interpretation,
the process of decimationis often referred as down sampling. The block diagram for this is shown in
figure (8.16)
Fig 8.16
There are situations in which it is useful to convert a sequence to a higher equivalent sampling rate.
This process is referred to as upsampling or interpolation. This process is reverse of the
downsampling. Given a sequence we obtain an expanded sequence by inserting (L -
1) zero.
(8.19)
The interpolated sequence is obtained by low pass filtering of
After low pass filtering
(8.20)
For ideal low-pass filter with cutoff and gain L we get
(8.21)
Signals and their spectra interpolation are shown in figure (8.17)
Fig 8.17
We can get a non integer change in rate if it is ratio of two integers by using upsampling and
downsampling operations.
In many applications of signal processing we want to change the relative amplitudes and frequency
contents of a signal. This process is generally referred to as filtering. Since the Fourier transform of
the output is product of input Fourier transform and frequency response of the system, we have to
use appropriate frequency response.
Ideal frequency selective filters
An ideal frequency reflective filter passes complex exponential signal. for a given set of frequencies
and completely rejects the others. Figure (9.1) shows frequency response for ideal low pass filter
(LPF), ideal high pass filter (HPF), ideal bandpass filter (BPF) and ideal backstop filter (BSF).
Fig 9.1
The ideal filters have a frequency response that is real and non-negative, in other words, has a zero
phase characteristics. A linear phase characteristics introduces a time shift and this causes no
distortion in the shape of the signal in the passband.
Since the Fourier transfer of a stable impulse response is continuous function of , can not get a
stable ideal filter.
Filter specification
Since the frequency response of the realizable filter should be a continuous function, the magnitude response of a
lowpass filter is specified with some acceptable tolerance. Moreover, a transition band is specified between the
passband and stop band to permit the magnitude to drop off smoothly. Figure (9.2) illustrates this
Fig 9.2
In the passband magnitude the frequency response is within of unity
In the stopband
The frequencies and are respectively, called the passband edge frequency and the stopband edge
frequency. The limits on tolerances and are called the peak ripple value. Often the specifications of digital
filter are given in terms of the loss function , in dB. The loss specification of digital filter
are
Some times the maximum value in the passband is assumed to be unity and the maximum passband deviation,
denoted as is given the minimum value of the magnitude in passband. The maximum stopband magnitude
is denoted by. The quantity is given by
These are illustrated in Fig(9.3)
Fig 9.3
If the phase response is not specified, one prefers to use IIR digital filter. In case of an IIR filter design, the most
common practice is to convert the digital filter specifications to analog low pass prototype filter specifications, to
determine the analog low pass transfer function meeting these specifications, and then to transform it into
desired digital filter transfer function. This methods is used for the following reasons:
1. Analog filter approximation techniques are highly advanced.
2. They usually yield closed form solutions.
3. Extensive tables are available for analog-design.
4. Many applications require the digital solutions of analog filters.
The transformations generally have two properties (1) the imaginary axis of the s-plane maps into unit circle of the
z-plane and (2) a stable continuous time filter is transformed to a stable discrete time filter.
Filter design by impulse invariance
In the impulse variance design procedure the impulse response of the impulse response of
the discrete time system is proportional to equally spaced samples of the continues time filter, i.e.,
where Td represents a sampling interval, since the specifications of the filter are given in discrete time
domain, it turns out that Td has no role to play in design of the filter. From the sampling theorem we
know that the frequency response of the discrete time filter is given by
Since any practical continuous time filter is not strictly bandlimited there issome aliasing. However, if
the continuous time filter approaches zero at high frequencies, the aliasing may be negligible. Then
the frequency response of the discrete time filter is
We first convert digital filter specifications to continuous time filter specifications. Neglecting aliasing,
we get specification by applying the relation
(9.2)
where is transferred to the designed filter H(z), we again use equation (9.2) and the
parameter Tdcancels out.
Let us assume that the poles of the continuous time filter are simple, then
The corresponding impulse response is
Then
The system function for this is
We see that a pole at in the s-plane is transformed to a pole at Td in the z-plane. If
the continuous time filter is stable, that is , then the magnitude of will be less
than 1, so the pole will be inside unit circle. Thus the causal discrete time filter is stable. The mapping
of zeros is not so straight forward.
Example:
Design a lowpass IIR digital filter H(z) with maximally flat magnitude characteristics. The passband
edge frequency is with a passband ripple not exceeding 0.5dB. The minimum stopband
attenuation at the stopband edge frequency of is 15 dB.
We assume that no aliasing occurs. Taking , the analog filter has ,
the passband ripple is 0.5dB, and minimum stopped attenuation is 15dB. For maximally flat frequency
response we choose Butterworth filter characteristics. From passband ripple of 0.5 dB we get
at passband edge.
From this we get
From minimum stopband attenuation of 15 dB we get
at stopped edge
The inverse discrimination ratio is given by
and inverse transition ratio is given by
Since N must be integer we get N=4. By we get
The normalized Butterworth transfer function of order 4 is given by
This is for normalized frequency of 1 rad/s. Replace s by to get , from this we get
Bilinear Transformation
This technique avoids the problem of aliasing by mapping axis in the s-plane to one revaluation of
the unit circle in the z-plane.
If is the continues time transfer function the discrete time transfer function is detained by
replacing s with
(9.3)
Rearranging terms in equation (9.3) we obtain.
Substituting , we get
If , it is then magnitude of the real part in denominator is more than that of the numerator and
so. Similarly if , than for all. Thus poles in the left half of the s-plane will get mapped to
the poles inside the unit circle in z-plane. If then
So, , writing we get
rearranging we get
or
(9.5)
or
(9.6)
The compression of frequency axis represented by (9.5) is nonlinear. This is illustrated in figure 9.4.
Fig 9.4
Because of the nonlinear compression of the frequency axis, there is considerable phase distortion in
the bilinear transformation.
Example
We use the specifications given in the previous example. Using equation (9.5) with we get
Some frequently used analog filters
In the previous two examples we have used Butterworth filter. The Butterworth filter of order n is
described by the magnitude square frequency response of
It has the following properties
1.
2.
3. is monotonically decreasing function of
4. As n gets larger, approaches an ideal low pass filter
5. is called maximally flat at origin, since all order derivative exist and they are zero
at
The poles of a Butterworth filter lie on circle of radius in s-plane.
There are two types of Chebyshev filters, one containing ripples in the passband (type I) and the other
containing a ripple in the stopband (type II). A Type I low pass normalizer Chebyshev filter has the
magnitude squared frequency response.
where is nth
order Chebyshev polynomial. We have the relationship
with
Chebyshev filters have the following properties
1. The magnitude squared frequency response oscillates between 1 and within the
passband, the so called equiripple and has a value of at , the normalized cut
off frequency.
2. The magnitude response is monotonic outside the passband including transitionand stopband.
3. The poles of the Chebysher filter lie on an ellipse in s-plane.
An elliptic filter has ripples both in passband and in stopband. The square magnitude frequency
response is given by
where is Chebyshev rational function of O determined from specified ripple characteristics.
An nth
order Chebyshev filter has sharper cutoff than a Butterworth filter, that is, has a narrower
transition bandwidth. Elliptic filter provides the smallest transition width.
Design of Digital filter using Digital to Digital transformation
There exists a set of transformation that takes a low pass digital filter and turn into highpass,
bandpass, bandstop or another lowpass digital filter. These transformations are given in table 9.1.
The transformations all take the form of replacing the in by some function of.
Type From To Transformation Design Formula
Low pass
cutoff
Low pass
cutoff
LPF HPF
LPF BPF
LPF BSF
Starting with a set of digital specifications and using the inverse of the design equation given in table
9.1, a set of lowpass digital requirements can be established. A LPF digital prototype filter is
then selected to satisfy these requirements and the proper digital to digital transformation is applied
to give the desired.
Example
Using the digital to digital transformation, find the system function for a low-pass digital filter
that satisfies the following set the requirements (a) monotone stop and passband (b)-3dB cutoff
frequency of (c) attenuation at and past is at least 15dB.
Because of monotone requirement, a Butterworth filter is selected. The required n is given by
rounded to 2.
For we get from table 9.1. , From standard tables (or MATLAB) we find
standard 2 nd order Butterworth filter with cut off and then apply the digital transform to get
FIR filter design
In the previous section, digital filters were designed to give a desired frequency response magnitude
without regard to the phase response. In many cases a linear phase characteristics is required through
the passband of the filter. It can be shown that causal IIR filter cannot produce a linear phase
characteristics and only special forms of causal FIR filters can give linear phase.
If represents the impulse response of a discrete time linear system a necessary and sufficient
condition for linear phase is that have finite duration N , that it be symmetric about its mid
point, i.e.
For N even, we get
For N odd
For N even we get a non-integer delay, which will cause the value of the sequenceto change, [See
continuous time implementation of discrete time system, for interpretation of non-integer delay].
One approach to design FIR filters with linear phase is to use windowing.
The easiest way to obtain an FIR filter is to simply truncate the impulse response of an IIR filter.
If is the impulse response of the designed FIR filter, then an FIR filter with
impulseresponse can be obtained as follows.
This can be thought of as being formed by a product of and a window function
where is said to be rectangular window and is given by
Using modulation property of Fourier transfer
For example if is ideal low pass filter and is rectangular window is measured version
of the ideal low pass frequency response.
Fig 9.5
In general, the index the main lobe of , the more spreading where as the narrower
the main lobe (larger N), the closer comes to. In general, we are left with a trade-off of
making N large-enough so that smearing is minimized, yet small enough to allow reasonable
implementation. Much work has been done on adjusting to satisfy certain main lobe and side
lobe requirements. Some of the commonly used windows are give in below.
(a) Rectangular
(b) Bartlett (or triangle)
(c) Hanning
(d) Harming
(e) Blackman
(f) Kaiser
where is modified zero-order Bessel function of the first kind given by
The main lobe width and first side lobe attenuation increase as we proceed down the window listed
above.
An ideal lowpass filter with linear phase and cut off is characterized by
The corresponding impulse response is
Since this is symmetric about , if we change and use one of the windows listed
above the will get linear phase FIR filter. Transition width and minimum stopped attenuation are listed
in the Table 9.3.
Window Transition Width Minimum stopband
attenuation
Rectangular
-21db
Bartlett
-25dB
Hanning
-44dB
Hamming
-53dB
Blackman
-74dB
Kaiser variable variable
Table 9.3
We first choose a window that satisfies the minimum attenuation. The transition bandwidth is
approximately that allows us to choose the value of N. Actual frequency response characteristic are
then calculated and we see if the requirements are met or not. Accordingly N is adjusted parameters
for kaiser window are obtained from design formula available for this MATLAB or similar programmes
have all there formulas.
Realizations of Digital Filters
We have many realizations of digital filter. Some of these are now discussed. Direct Form Realization -
An important class of linear time -invariant systems is characterized by the transfer function.
A system with input and output could be realized by the following constant coefficient
difference equation
A realization of the filter using equation (9.31) is shown in figure (9.6)
Fig 9.6 Direct form I
The output is seen to be weighted sum of input and past inputs and
past outputs. Another realization can be obtained by uniting as product of two transfer
functions and , where contains only the denominator or poles and
contains only the numerator or zeros as follows
where
Fig 9.7
The output of the filter is obtained by calculating the intermediate result obtained from
operating on the input with filter and then operating on w[n] with filter.Thus we obtain
or
and
or
The realization is shown in figure 9.8
Fig 9.8
Upon close examination of Fig 9.8, it can be seen that the two branches of delay elements can be
combined as they both refer to delayed versions of and upon simplification, the direct form II
canonical realization is obtained as shown in figure 9.9.
Fig 9.9 Direct form II
In this form the number of delay element is max (M,N). It can be shown that this is the minimum
number of delay elements that are required to implement the digital filter. This does not mean that
this is the best realization. Immunity to roundoff and quantization are very important considerations.
An important special case that is used as building block occurs when. Thus is ratio of two
qualities in , called biquadratic section, and is given by
The alternative form is found to be useful for amplitude scaling for improving performance file filter
operation. This form is shown in figure 9.10.
Fig 9.10
Cascade Realizations: In the cascade realization is broken into productof transfer
functions each a rational expression in as follows
Fig 9.11
could be broken up in many ways; however the most common method is to use biquadratic
sections. Thus
by letting and equal to zero we get bilinear section. Even among the biquadratic sections
we have many choices as how we pair poles and zeros. Also the order of the sections can be different
Example:
Final cascade realization of
Using only real coefficients can be decompressed as
Divides both numerator and denominator by and factoring 8 as , one possible
rearrangement for is
This can be realized as shown is figure 9.12
Fig 9.12
Parallel Realizations:
The transfer function H ( z ) could be written as a sum of transfer functions
as follows:
One parallel form results when are all selected to be of the following form for
If , we will have a section of FIR filter, obtained by performing long division. Once
denominator polynomial has degree more than the numerator polynomial we perform the partial
fraction expansion. The resulting structure is shown in figure 9.13.
Fig 9.13
Example:Find the parallel form for the filter given in last example.
Using MATLAB program or otherwise we get
using direst form realization for individual section we get the structure shown in figure 9.14.
Fig 9.14 Apart from these there exist a number of other realizations like lattice form, state variable
realization etc.